Tehtävät
https://peda.net/id/adf6b704de8
2019-09-24T08:53:49+03:00
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Teksti
https://peda.net/id/5a06cb1603c
2019-11-10T16:49:52+02:00
370<br/>
<div> Läpipääsyyn vaaditaan 15 oikeaa vastausta, ja opiskelija tietää 10 vastausta, joten hän pääsee läpi arvaamalla oikein 5 vastausta lopuista 15:sta.</div>
<div> Koska opiskelija arvaa vastauksen 15 tehtävään, joissa vastausvaihtoehtoja on kaksi, tilannetta voidaan ajatella 15 yrityksen toistokokeena, jossa onnistumistodennäköisyys on jokaisella toistolla 1/2.</div>
<div> Opiskelija tarvitsee läpipääsyyn tietämiensä 10 vastauksen lisäksi vähintään 5 oikein arvattua vastausta eli 5, 6, … tai 10 oikeaa arvausta. Lasketaan vastatapahtuman "korkeintaan 4 oikeaa arvausta" eli "0, 1, 2, 3 tai 4 oikeaa arvausta" todennäköisyys. Tapahtumat ovat erilliset oikeiden vastausten eri lukumäärille, joten yhteenlaskusäännöllä saadaan </div>
<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(0%7B%2C%7D1%7B%2C%7D3%5C%20tai%5C%204%5C%20arvausta%5C%20oikein%5Cright)" alt="P\left(0{,}1{,}3\ tai\ 4\ arvausta\ oikein\right)"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3DP%5Cleft(0%5C%20oikein%5Cright)%2BP%5Cleft(1%5C%20oikein%5Cright)%2BP%5Cleft(2%5C%20oikein%5Cright)%2BP%5Cleft(3%5C%20oikein%5Cright)%2BP%5Cleft(4%5C%20oikein%5Cright)" alt="=P\left(0\ oikein\right)+P\left(1\ oikein\right)+P\left(2\ oikein\right)+P\left(3\ oikein\right)+P\left(4\ oikein\right)"/><br/>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(1-%5Cfrac%7B1%7D%7B2%7D%5Cright)%5E%7B15%7D%2B%5Cbinom%7B15%7D%7B1%7D%5Ccdot%5C%20%5Cleft(1-%5Cfrac%7B1%7D%7B2%7D%5Cright)%5E%7B15-1%7D%2B%5Cbinom%7B15%7D%7B2%7D%5Ccdot%5Cleft(%5Cfrac%7B1%7D%7B2%7D%5Cright)%5E2%5Ccdot%5Cleft(1-%5Cfrac%7B1%7D%7B2%7D%5Cright)%5E%7B15-2%7D%2B%5Cbinom%7B15%7D%7B3%7D%5Ccdot%5Cleft(%5Cfrac%7B1%7D%7B2%7D%5Cright)%5E3%5Ccdot%5Cleft(1-%5Cfrac%7B1%7D%7B2%7D%5Cright)%5E%7B15-5%7D%2B%5Cbinom%7B15%7D%7B4%7D%5Ccdot%5Cleft(%5Cfrac%7B1%7D%7B2%7D%5Cright)%5E4%5Ccdot%5Cleft(1-%5Cfrac%7B1%7D%7B2%7D%5Cright)%5E%7B15-4%7D" alt="=\left(1-\frac{1}{2}\right)^{15}+\binom{15}{1}\cdot\ \left(1-\frac{1}{2}\right)^{15-1}+\binom{15}{2}\cdot\left(\frac{1}{2}\right)^2\cdot\left(1-\frac{1}{2}\right)^{15-2}+\binom{15}{3}\cdot\left(\frac{1}{2}\right)^3\cdot\left(1-\frac{1}{2}\right)^{15-5}+\binom{15}{4}\cdot\left(\frac{1}{2}\right)^4\cdot\left(1-\frac{1}{2}\right)^{15-4}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3D0%7B%2C%7D05892..." alt="=0{,}05892..."/></div>
<span> Siis todennäköisyys, että opiskelija läpäisee testin, on 1-P(korkeintaan 4 arvausta oikein)=1-0,05892...=0,9410...≈0,94. <br/>
</span><br/>
371<br/>
<span>a) </span>
<div>Ajatellaan tilannetta toistokokeena, jossa onnistumisen todennäköisyys on P(O) = 0,33 ja toistoja on 12. Tapahtuman "enintään 9 O:ta" vastatapahtuma on "vähintään 10 O:ta" eli "10, 11 tai 12 O:ta". Erillisten tapahtumien yhteenlaskusäännöllä saadaan <br/>
<br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(10%7B%2C%7D%5C%2011%5C%20tai%5C%2012%5C%20O%3Ata%5Cright)" alt="P\left(10{,}\ 11\ tai\ 12\ O:ta\right)"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3DP%5Cleft(10%5C%20O%3Ata%5Cright)%2BP%5Cleft(11%5C%20O%3Ata%5Cright)%2BP%5Cleft(12%5C%20O%3Ata%5Cright)" alt="=P\left(10\ O:ta\right)+P\left(11\ O:ta\right)+P\left(12\ O:ta\right)"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cbinom%7B12%7D%7B10%7D%5Ccdot0.33%5E%7B10%7D%5Ccdot%5Cleft(1-0%7B%2C%7D33%5Cright)%5E%7B12-10%7D%2B%5Cbinom%7B12%7D%7B11%7D%5Ccdot0%7B%2C%7D33%5E%7B11%7D%5Ccdot%5Cleft(1-0%7B%2C%7D33%5Cright)%5E%7B12-11%7D%2B0%7B%2C%7D33%5E%7B12%7D" alt="=\binom{12}{10}\cdot0.33^{10}\cdot\left(1-0{,}33\right)^{12-10}+\binom{12}{11}\cdot0{,}33^{11}\cdot\left(1-0{,}33\right)^{12-11}+0{,}33^{12}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3D0%7B%2C%7D0004960..." alt="=0{,}0004960..."/>
<div>Siispä</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(enit%C3%A4%C3%A4n%5C%209%5C%20O%3Ata%5Cright)" alt="P\left(enitään\ 9\ O:ta\right)"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3D1-P%5Cleft(enit%C3%A4%C3%A4n%5C%2010%5C%20O%3Ata%5Cright)" alt="=1-P\left(enitään\ 10\ O:ta\right)"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3D1-0%7B%2C%7D0004960..." alt="=1-0{,}0004960..."/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3D0%7B%2C%7D99950" alt="=0{,}99950"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Capprox0%7B%2C%7D9995" alt="\approx0{,}9995"/><br/>
<div>b)</div>
<div> Nyt toistokokeen onnistumistodennäköisyys on P(B) = 0,17. Tapahtumat "joukossa on kolme B:tä" ja "joukossa on "neljä B:tä" ovat erilliset, joten yhteenlaskusäännöllä saadaan </div>
<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(3%5C%20tai%5C%204%5C%20B%3At%C3%A4%5Cright)" alt="P\left(3\ tai\ 4\ B:tä\right)"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3DP%5Cleft(3%5C%20B%3At%C3%A4%5Cright)%2BP%5Cleft(4%5C%20B%3At%C3%A4%5Cright)" alt="=P\left(3\ B:tä\right)+P\left(4\ B:tä\right)"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cbinom%7B12%7D%7B3%7D%5Ccdot0%7B%2C%7D17%5E3%5Ccdot%5Cleft(1-0%7B%2C%7D17%5Cright)%5E%7B12-3%7D%2B%5Cbinom%7B12%7D%7B4%7D%5Ccdot0%7B%2C%7D17%5E4%5Ccdot%5Cleft(1-0%7B%2C%7D17%5Cright)%5E%7B12-4%7D" alt="=\binom{12}{3}\cdot0{,}17^3\cdot\left(1-0{,}17\right)^{12-3}+\binom{12}{4}\cdot0{,}17^4\cdot\left(1-0{,}17\right)^{12-4}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3D0%7B%2C%7D295" alt="=0{,}295"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Capprox0%7B%2C%7D30" alt="\approx0{,}30"/></div>
</div>
<br/>
383
<div>Tapahtuman "ainakin kaksi itää" vastatapahtuma on "korkeintaan yksi itää" eli "ei yhtään tai yksi itää". Lasketaan vastatapahtuman todennäköisyys. </div>
<div>Ajatellaan tilannetta toistokokeena, jossa onnistumisen todennäköisyys on P(sipuli itää) = 0,7. </div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(enit%C3%A4%C3%A4n%5C%20yksi%5C%20it%C3%A4%C3%A4%5Cright)%3DP%5Cleft(ei%5C%20yht%C3%A4%C3%A4n%5Cright)%2BP%5Cleft(yksi%5Cright)" alt="P\left(enitään\ yksi\ itää\right)=P\left(ei\ yhtään\right)+P\left(yksi\right)"/></div>
<br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(1-0%7B%2C%7D7%5Cright)%5En%2B%5Cbinom%7Bn%7D%7B1%7D%5Ccdot0%7B%2C%7D7%5E1%5Ccdot%5Cleft(1-0%7B%2C%7D7%5Cright)%5E%7Bn-1%7D" alt="=\left(1-0{,}7\right)^n+\binom{n}{1}\cdot0{,}7^1\cdot\left(1-0{,}7\right)^{n-1}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3D0%7B%2C%7D3%5En%2Bn%5Ccdot0%7B%2C%7D7%5Ccdot0%7B%2C%7D3%5E%7Bn-1%7D" alt="=0{,}3^n+n\cdot0{,}7\cdot0{,}3^{n-1}"/>
<div>Siis <br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(ainakin%5C%20kaksi%5C%20it%C3%A4%C3%A4%5Cright)%3D1-%5Cleft(0%7B%2C%7D3%5En%2Bn%5Ccdot0%7B%2C%7D7%5Ccdot0%7B%2C%7D3%5E%7Bn-1%7D%5Cright)" alt="P\left(ainakin\ kaksi\ itää\right)=1-\left(0{,}3^n+n\cdot0{,}7\cdot0{,}3^{n-1}\right)"/></div>
<div>Etsitään siis pienin luonnollinen luku n, jolle<span> </span><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(ainakin%5C%20kaksi%5C%20it%C3%A4%C3%A4%5Cright)%5Cge0%7B%2C%7D99" alt="P\left(ainakin\ kaksi\ itää\right)\ge0{,}99"/>eli<img src="https://math-demo.abitti.fi/math.svg?latex=1-%5Cleft(0%7B%2C%7D3%5En%2Bn%5Ccdot0%7B%2C%7D7%5Ccdot0%7B%2C%7D3%5E%7Bn-1%7D%5Cright)%5Cge0%7B%2C%7D99" alt="1-\left(0{,}3^n+n\cdot0{,}7\cdot0{,}3^{n-1}\right)\ge0{,}99"/></div>
<div>Etsitään luku n ratkaisemalla ensin yhtälö <img src="https://math-demo.abitti.fi/math.svg?latex=1-%5Cleft(0%7B%2C%7D3%5En%2Bn%5Ccdot0%7B%2C%7D7%5Ccdot0%7B%2C%7D3%5E%7Bn-1%7D%5Cright)%3D0%7B%2C%7D99" alt="1-\left(0{,}3^n+n\cdot0{,}7\cdot0{,}3^{n-1}\right)=0{,}99"/>symbolisen laskennan ohjelmalla</div>
<div>Saadaan n=6,05...</div>
<div><br/>
<div>Saadaan n = 6,05… Mitä enemmän sipuleita istutetaan, sitä suurempi on todennäköisyys, että vähintään haluttu määrä sipuleita itää. Sipuleita on siis istutettava vähintään 7. </div>
</div>
2019-11-10T16:49:52+02:00
Teksti
https://peda.net/id/6a049c66fe7
2019-11-03T22:56:31+02:00
<span>4.4 Normaalijakauma</span>
<div> </div>
<div>Esimerkki. Mailan kestoikä noudattaa normaalijakaumaa.Kestoiän odotusarvo on 960 peliminuutttia ja keskihajonta 105 min. Millä todennäköisyydellä maila hajoaa 870 ja 1000 min välillä?</div>
<div> </div>
<div>Ratkaisu </div>
<div>Tapa1:</div>
<div>X= Mailan kestoikä (min)</div>
<div>X~N(960,105)</div>
<div>P(870 ≤ X ≤1000)</div>
<div> </div>
<div>Normitetaan satunaismuuttuja X.</div>
<img src="https://math-demo.abitti.fi/math.svg?latex=Z%3D%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%3D%5Cfrac%7BX-960%7D%7B105%7D" alt="Z=\frac{X-\mu}{\sigma}=\frac{X-960}{105}"/>
<div>Kun X=1000, niin</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1000-960%7D%7B105%7D%3D%5Cfrac%7B40%7D%7B105%7D%5Capprox0%7B%2C%7D38" alt="\frac{1000-960}{105}=\frac{40}{105}\approx0{,}38"/></div>
<div>Kun X=870</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B870-960%7D%7B105%7D%3D-%5Cfrac%7B10%7D%7B105%7D%5Capprox-0%7B%2C%7D86" alt="\frac{870-960}{105}=-\frac{10}{105}\approx-0{,}86"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(870%5Cle%20X%5Cle1000%5Cright)%3DP%5Cleft(-0%7B%2C%7D86%5Cle%20Z%5Cle0%7B%2C%7D38%5Cright)" alt="P\left(870\le X\le1000\right)=P\left(-0{,}86\le Z\le0{,}38\right)"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3DP%5Cleft(Z%5Cle0%7B%2C%7D38%5Cright)-P%5Cleft(Z%5Cle-0%7B%2C%7D86%5Cright)%3D%5CPhi%5Cleft(0%7B%2C%7D38%5Cright)-%5CPhi%5Cleft(-0%7B%2C%7D86%5Cright)" alt="=P\left(Z\le0{,}38\right)-P\left(Z\le-0{,}86\right)=\Phi\left(0{,}38\right)-\Phi\left(-0{,}86\right)"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5CPhi%5Cleft(0%7B%2C%7D38%5Cright)-%5Cleft(1-%5CPhi%5Cleft(0%7B%2C%7D86%5Cright)%5Cright)" alt="=\Phi\left(0{,}38\right)-\left(1-\Phi\left(0{,}86\right)\right)"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5CPhi%5Cleft(0%7B%2C%7D38%5Cright)-1%2B%5CPhi%5Cleft(0%7B%2C%7D86%5Cright)" alt="=\Phi\left(0{,}38\right)-1+\Phi\left(0{,}86\right)"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3D0%7B%2C%7D6480-1%2B0%7B%2C%7D8051" alt="=0{,}6480-1+0{,}8051"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3D0%7B%2C%7D4531%5Capprox0%7B%2C%7D45%3D45%5C%25" alt="=0{,}4531\approx0{,}45=45\%"/><br/>
<div> </div>
</div>
<div>Tapa 2:</div>
<div>Todennäköisyyslaskurilla saadaan kysytyksi todennäköisyydeksi 0,4527 eli noin 45%</div>
<div> </div>
<div>481</div>
<div>X= betonisäkin massa (kg)</div>
<div>X~N(µ; 0,9)</div>
<div>Mikä on odotusarvo µ, kun P(X>25)= 0,95 eli P(X≤25)=0,05?</div>
<div> </div>
<div>Ratkaistaan geogebran CAS-laskimella</div>
<div>µ≈26,48</div>
<div> </div>
<div>463</div>
<div>a) 0,1587 ≈ 16%</div>
<div>b) 0,1587 ≈ 16%</div>
<div>c) 0,6827 ≈ 67%</div>
<div> </div>
<div>466</div>
<div>a) 16973,8 ≈ 17000</div>
<div>b) 14695,9 ≈ 14700 </div>
<div>c) 17464,4 ≈ 17500</div>
<div> </div>
<div>468</div>
<div>a) C 0,1587</div>
<div>b) A 0,2266</div>
<div>c) B 0,1587</div>
<div>d) D 0,0668</div>
<div> </div>
<div>469</div>
<div>471</div>
<div>475</div>
<div>477</div>
<div>483</div>
2019-11-03T22:56:31+02:00
Teksti
https://peda.net/id/3d506f08fa1
2019-11-14T10:48:58+02:00
<span><br/>
442<br/>
a) Funktio f ei ole tiheysfunktio, koska sen kuvaajan ja x-akselin väliin<br/>
jäävän alueen pinta-ala ei ole 1 (vaan 2).<br/>
Funktio g ei ole tiheysfunktio, koska sen kuvaajan ja x-akselin väliin<br/>
jäävän alueen pinta-ala ei ole 1 (vaan suurempi).<br/>
Funktio h ei ole tiheysfunktio, koska se saa negatiivisia arvoja.<br/>
b) Funktio k on tiheysfunktio, koska se ei saa missään negatiivisia arvoja<br/>
ja lisäksi sen kuvaajan ja x-akselin väliin jäävän alueen pinta-ala on 1. <br/>
<br/>
444</span>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%3Da%5Cright)%3D0%7B%2C%7D37" alt="P\left(X=a\right)=0{,}37"/></div>
<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(a%5Cle%20X%5Cle%20b%5Cright)%3D0%7B%2C%7D39" alt="P\left(a\le X\le b\right)=0{,}39"/><br/>
<div>a)<br/>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%5Cle%20a%5Cright)%3DP%5Cleft(X%5Cle%20a%5Cright)-P%5Cleft(X%3Da%5Cright)" alt="P\left(X\le a\right)=P\left(X\le a\right)-P\left(X=a\right)"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%3Da%5Cright)" alt="P\left(X=a\right)"/>tarkoittaa funktion kuvaajan ja x-akselin väliin jäävä pinta-ala kohdassa x=a eli janan pinta-alaa. Janan pinta-ala on nolla eli jakautuneen satunnaismuuttujan yksittäisen arvon todennäköisyys on nolla eli </div>
<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%3Da%5Cright)%3D0" alt="P\left(X=a\right)=0"/>. Siten<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%5Cle%20a%5Cright)%3DP%5Cleft(X%3Ca%5Cright)%3D0%7B%2C%7D37" alt="P\left(X\le a\right)=P\left(X<a\right)=0{,}37"/>.</div>
<div><br/>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%3Cb%5Cright)%3DP%5Cleft(X%5Cle%20b%5Cright)-P%5Cleft(X%3Db%5Cright)%3DP%5Cleft(X%3Ca%5Cright)%2BP%5Cleft(a%5Cle%20X%5Cle%20b%5Cright)%3D0%7B%2C%7D37%2B0%7B%2C%7D39%3D0%7B%2C%7D76" alt="P\left(X<b\right)=P\left(X\le b\right)-P\left(X=b\right)=P\left(X<a\right)+P\left(a\le X\le b\right)=0{,}37+0{,}39=0{,}76"/></div>
<div> </div>
<div>c)Tapahtumat X > a ja X ≤ a ovat oistensa vastatapahtumia, joten</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%3Ea%5Cright)%3D1-P%5Cleft(X%5Cle%20a%5Cright)%3D1-0%7B%2C%7D37%3D0%7B%2C%7D63" alt="P\left(X>a\right)=1-P\left(X\le a\right)=1-0{,}37=0{,}63"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"--></div>
<div> </div>
<div>d)Tapahtumat X ≥ b ja X < b ovatvastatapahtumia, joten</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%5Cge%20b%5Cright)%3D1-P%5Cleft(X%3Cb%5Cright)%3D1-0%7B%2C%7D76%3D0%7B%2C%7D24" alt="P\left(X\ge b\right)=1-P\left(X<b\right)=1-0{,}76=0{,}24"/></div>
<br/>
<div>445<br/>
Kuvassa A on funktion f kuvaajan ja x-akselin väliin välillä a ≤ x ≤ b<br/>
jäävän alueen pinta-ala. Kun f on satunnaismuuttujan X tiheysfunktio,<br/>
tämä pinta-ala on sama kuin todennäköisyys P(a ≤ X ≤ b), joka on sama<br/>
kuin P(a < X ≤ b). Siis kuvaan A sopivat merkinnät II ja VI.<br/>
Kuvassa B on funktion f kuvaajan ja x-akselin väliin välillä x > b jäävän<br/>
alueen pinta-ala, joka on sama kuin todennäköisyys P(X > b) eli merkintä V.<br/>
Kuvassa C on funktion f kuvaajan ja x-akselin väliin välillä x < b jäävän<br/>
alueen pinta-ala, joka on sama kuin todennäköisyys P(X ≤ b) eli merkintä I.<br/>
Kuvassa D on funktion f kuvaajan ja x-akselin väliin väleillä x < a ja x > b<br/>
jäävän kaksiosaisen alueen pinta-ala, joka on sama kuin todennäköisyys<br/>
P(X < a tai X > b). Tapahtuman "X < a tai X > b" vastatapahtuma on<br/>
tapahtuma a ≤ X ≤ b, jonka todennäköisyys näkyy kuvassa valkoisen<br/>
alueen pinta-alana. Siis kuvaan merkittyä (väritettyä) pinta-alaa vastaa<br/>
todennäköisyys P(X < a tai X > b) = 1 − P(a ≤ X ≤ b) eli merkintä III. </div>
<br/>
<br/>
<br/>
</div>
2019-10-29T09:03:41+02:00
Teksti
https://peda.net/id/52bcd458fa1
2019-10-29T09:04:17+02:00
422
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%3D0%5Cright)%3D%5Cfrac%7B3%7D%7B8%7D" alt="P\left(X=0\right)=\frac{3}{8}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%3D1%5Cright)%3D%5Cfrac%7B1%7D%7B4%7D" alt="P\left(X=1\right)=\frac{1}{4}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%3D2%5Cright)%3D%5Cfrac%7B1%7D%7B8%7D" alt="P\left(X=2\right)=\frac{1}{8}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%3D3%5Cright)%3D%5Cfrac%7B1%7D%7B4%7D" alt="P\left(X=3\right)=\frac{1}{4}"/> <br/>
<div>a)</div>
<div> </div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=E%5Cleft(X%5Cright)%3D%5Cfrac%7B3%7D%7B8%7D%5Ccdot0%2B%5Cfrac%7B1%7D%7B4%7D%5Ccdot1%2B%5Cfrac%7B1%7D%7B8%7D%5Ccdot2%2B%5Cfrac%7B1%7D%7B4%7D%5Ccdot3%3D1%7B%2C%7D25" alt="E\left(X\right)=\frac{3}{8}\cdot0+\frac{1}{4}\cdot1+\frac{1}{8}\cdot2+\frac{1}{4}\cdot3=1{,}25"/></div>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(X%5Cright)%3D%5Csqrt%5B%5D%7B%5Cfrac%7B3%7D%7B8%7D%5Cleft(0-2%7B%2C%7D25%5Cright)%5E2%2B%5Cfrac%7B1%7D%7B4%7D%5Cleft(1-2%7B%2C%7D25%5Cright)%5E2%2B%5Cfrac%7B1%7D%7B8%7D%5Cleft(2-2%7B%2C%7D25%5Cright)%5E2%2B%5Cfrac%7B1%7D%7B4%7D%5Cleft(4-2%7B%2C%7D25%5Cright)%5E2%7D%3D1%7B%2C%7D198...%5Capprox1%7B%2C%7D20" alt="D\left(X\right)=\sqrt[]{\frac{3}{8}\left(0-2{,}25\right)^2+\frac{1}{4}\left(1-2{,}25\right)^2+\frac{1}{8}\left(2-2{,}25\right)^2+\frac{1}{4}\left(4-2{,}25\right)^2}=1{,}198...\approx1{,}20"/></div>
<div>
<div> </div>
<div>423</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0AKruuna%26x%26x%5C%5C%0A%5Chline%0AKlaava%26x%26x%5C%5C%0A%26Klaava%26Kruuna%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l} Kruuna&x&x\\ \hline Klaava&x&x\\ &Klaava&Kruuna \end{array}"/></div>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%26%5C%5C%0A%5Chline%0A2%262%5C%5C%0A1%26-1%5C%5C%0A0%260%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l} &\\ \hline 2&2\\ 1&-1\\ 0&0 \end{array}"/></div>
<div>a)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%3D0%5Cright)%3DP%5Cleft(2%5C%20klaavaa%5Cright)%3D%5Cfrac%7B1%7D%7B4%7D" alt="P\left(X=0\right)=P\left(2\ klaavaa\right)=\frac{1}{4}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%3D-1%5Cright)%3DP%5Cleft(1%5C%20kruuna%5Cright)%3D%5Cfrac%7B1%7D%7B2%7D" alt="P\left(X=-1\right)=P\left(1\ kruuna\right)=\frac{1}{2}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%3D2%5Cright)%3DP%5Cleft(2%5C%20kruunaa%5Cright)%3D%5Cfrac%7B1%7D%7B4%7D" alt="P\left(X=2\right)=P\left(2\ kruunaa\right)=\frac{1}{4}"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=E%5Cleft(X%5Cright)%3D0%5Ccdot%5Cfrac%7B1%7D%7B4%7D%2B%5Cleft(-1%5Cright)%5Ccdot%5Cfrac%7B1%7D%7B2%7D%2B%5Ccdot2%5Ccdot%5Cfrac%7B1%7D%7B4%7D%3D0" alt="E\left(X\right)=0\cdot\frac{1}{4}+\left(-1\right)\cdot\frac{1}{2}+\cdot2\cdot\frac{1}{4}=0"/></div>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=0%5C%20euroa" alt="0\ euroa"/><br/>
<div> </div>
<div>
<div>424</div>
<div>a)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0AVoitto%26Kpl%26Todenn%C3%A4k%C3%B6isyys%5C%5C%0A%5Chline%0A999%2650%26%5Cfrac%7B50%7D%7B3%5C%20000%5C%20000%7D%5C%5C%0A99%26100%26%5Cfrac%7B100%7D%7B3%5C%20000%5C%20000%7D%5C%5C%0A19%2630%5C%20000%26%5Cfrac%7B30%5C%20000%7D%7B3%5C%20000%5C%20000%7D%5C%5C%0A1%26300%5C%20000%26%5Cfrac%7B300%5C%20000%7D%7B3%5C%20000%5C%20000%7D%5C%5C%0A0%26420%5C%20000%26%5Cfrac%7B420%5C%20000%7D%7B3%5C%20000%5C%20000%7D%5C%5C%0A-1%263%5C%20000%5C%20000-750%5C%20150%3D2249850%26%5Cfrac%7B2%5C%20249%5C%20850%7D%7B3%5C%20000%5C%20000%7D%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l} Voitto&Kpl&Todennäköisyys\\ \hline 999&50&\frac{50}{3\ 000\ 000}\\ 99&100&\frac{100}{3\ 000\ 000}\\ 19&30\ 000&\frac{30\ 000}{3\ 000\ 000}\\ 1&300\ 000&\frac{300\ 000}{3\ 000\ 000}\\ 0&420\ 000&\frac{420\ 000}{3\ 000\ 000}\\ -1&3\ 000\ 000-750\ 150=2249850&\frac{2\ 249\ 850}{3\ 000\ 000} \end{array}"/></div>
<img src="https://math-demo.abitti.fi/math.svg?latex=E%5Cleft(X%5Cright)%3D%5Cfrac%7B50%7D%7B3%5C%20000%5C%20000%7D%5Ccdot999%2B%5Cfrac%7B100%7D%7B3%5C%20000%5C%20000%7D%5Ccdot99%2B%5Cfrac%7B30%5C%20000%7D%7B3%5C%20000%5C%20000%7D%5Ccdot19%2B%5Cfrac%7B300%5C%20000%7D%7B3%5C%20000%5C%20000%7D%5Ccdot1%2B%5Cfrac%7B420%5C%20000%7D%7B3%5C%20000%5C%20000%7D%5Ccdot0%2B%5Cfrac%7B2%5C%20249%5C%20850%7D%7B3%5C%20000%5C%20000%7D%5Ccdot-1%3D-0%7B%2C%7D44" alt="E\left(X\right)=\frac{50}{3\ 000\ 000}\cdot999+\frac{100}{3\ 000\ 000}\cdot99+\frac{30\ 000}{3\ 000\ 000}\cdot19+\frac{300\ 000}{3\ 000\ 000}\cdot1+\frac{420\ 000}{3\ 000\ 000}\cdot0+\frac{2\ 249\ 850}{3\ 000\ 000}\cdot-1=-0{,}44"/><br/>
<div>b)</div>
<div>Koska on 2 249 850 kpl arpaa jolla ei saa penniäkään, joten jos osetetaan 2 249 851 arpaa, ainakin yhdellä arvalla saa jonkin voiton.</div>
<div> </div>
<div>425</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bmatrix%7D%0A%5Cbegin%7Bmatrix%7D%0A6%267%268%269%2610%5C%5C%0A5%266%267%268%269%5C%5C%0A4%265%266%267%268%5C%5C%0A3%264%265%266%267%5C%5C%0A2%263%264%265%266%5C%5C%0A1%262%263%264%265%5C%5C%0A%261%262%263%264%0A%5Cend%7Bmatrix%7D%0A%5Cend%7Bmatrix%7D" alt="\begin{matrix} \begin{matrix} 6&7&8&9&10\\ 5&6&7&8&9\\ 4&5&6&7&8\\ 3&4&5&6&7\\ 2&3&4&5&6\\ 1&2&3&4&5\\ &1&2&3&4 \end{matrix} \end{matrix}"/></div>
<div>a)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0Ax%26P%5Cleft(X%3Dx%5Cright)%5C%5C%0A%5Chline%0A2%26%5Cfrac%7B1%7D%7B24%7D%5C%5C%0A3%26%5Cfrac%7B1%7D%7B12%7D%5C%5C%0A4%26%5Cfrac%7B1%7D%7B8%7D%5C%5C%0A5%26%5Cfrac%7B1%7D%7B6%7D%5C%5C%0A6%26%5Cfrac%7B1%7D%7B6%7D%5C%5C%0A7%26%5Cfrac%7B1%7D%7B6%7D%5C%5C%0A8%26%5Cfrac%7B1%7D%7B8%7D%5C%5C%0A9%26%5Cfrac%7B1%7D%7B12%7D%5C%5C%0A10%26%5Cfrac%7B1%7D%7B24%7D%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l} x&P\left(X=x\right)\\ \hline 2&\frac{1}{24}\\ 3&\frac{1}{12}\\ 4&\frac{1}{8}\\ 5&\frac{1}{6}\\ 6&\frac{1}{6}\\ 7&\frac{1}{6}\\ 8&\frac{1}{8}\\ 9&\frac{1}{12}\\ 10&\frac{1}{24} \end{array}"/></div>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctext%7BE%7D%5Cleft(X%5Cright)%3D2%5Ccdot%5Cfrac%7B1%7D%7B24%7D%2B3%5Ccdot%5Cfrac%7B1%7D%7B12%7D%2B4%5Ccdot%5Cfrac%7B1%7D%7B8%7D%2B5%5Ccdot%5Cfrac%7B1%7D%7B6%7D%2B6%5Ccdot%5Cfrac%7B1%7D%7B6%7D%2B7%5Ccdot%5Cfrac%7B1%7D%7B6%7D%2B8%5Ccdot%5Cfrac%7B1%7D%7B8%7D%2B9%5Ccdot%5Cfrac%7B1%7D%7B12%7D%2B10%5Ccdot%5Cfrac%7B1%7D%7B24%7D%3D6" alt="\text{E}\left(X\right)=2\cdot\frac{1}{24}+3\cdot\frac{1}{12}+4\cdot\frac{1}{8}+5\cdot\frac{1}{6}+6\cdot\frac{1}{6}+7\cdot\frac{1}{6}+8\cdot\frac{1}{8}+9\cdot\frac{1}{12}+10\cdot\frac{1}{24}=6"/></div>
<div> </div>
<div>427</div>
<div>a)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BP%7D%3DKukaan%5C%20ei%5C%20tee%5C%20mallia" alt="\overline{P}=Kukaan\ ei\ tee\ mallia"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BP%7D_1%3D1-P_1%3D1-0%7B%2C%7D65%3D0%7B%2C%7D35" alt="\overline{P}_1=1-P_1=1-0{,}65=0{,}35"/></div>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BP%7D_2%3D1-P_2%3D1-0%7B%2C%7D75%3D0%7B%2C%7D25" alt="\overline{P}_2=1-P_2=1-0{,}75=0{,}25"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BP%7D_3%3D1-P_3%3D1-0%7B%2C%7D54%3D0%7B%2C%7D46" alt="\overline{P}_3=1-P_3=1-0{,}54=0{,}46"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BP%7D%3D%5Coverline%7BP%7D_1%5Ccdot%5Coverline%7BP%7D_2%5Ccdot%5Coverline%7BP%7D_3%3D0%7B%2C%7D04025" alt="\overline{P}=\overline{P}_1\cdot\overline{P}_2\cdot\overline{P}_3=0{,}04025"/></div>
Siten</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%3DAinakin%5C%20yksi%5C%20tekee%5C%20mallia" alt="P=Ainakin\ yksi\ tekee\ mallia"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%3D1-%5Coverline%7BP%7D%3D0%7B%2C%7D9597...%5Capprox0%7B%2C%7D96" alt="P=1-\overline{P}=0{,}9597...\approx0{,}96"/><br/>
<div>
<div>b)</div>
<img src="https://math-demo.abitti.fi/math.svg?latex=E%5Cleft(x%5Cright)%3D1%5Ccdot0%7B%2C%7D65%2B1%5Ccdot0%7B%2C%7D75%2B1%5Ccdot0%7B%2C%7D54%3D1%7B%2C%7D94" alt="E\left(x\right)=1\cdot0{,}65+1\cdot0{,}75+1\cdot0{,}54=1{,}94"/></div>
<div><br/>
<div>428</div>
<div>5 punaista</div>
<div>3 vihreää</div>
<div>1 musta</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0ATulos%26Voitto%5C%5C%0A%5Chline%0A2%5C%20samaa%265%5C%5C%0APunainen%5C%20JA%5C%20Vihre%C3%A4%261%5C%5C%0AAinakin%5C%20yksi%5C%20on%5C%20musta%26-5%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l} Tulos&Voitto\\ \hline 2\ samaa&5\\ Punainen\ JA\ Vihreä&1\\ Ainakin\ yksi\ on\ musta&-5 \end{array}"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0AYht.%269%5C%5C%0A%5Chline%0APunainen%26%5Cfrac%7B5%7D%7B9%7D%5C%5C%0AVihre%C3%A4%26%5Cfrac%7B1%7D%7B3%7D%5C%5C%0AMusta%26%5Cfrac%7B1%7D%7B9%7D%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l} Yht.&9\\ \hline Punainen&\frac{5}{9}\\ Vihreä&\frac{1}{3}\\ Musta&\frac{1}{9} \end{array}"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0Ax%26P%5Cleft(X%3Dx%5Cright)%5C%5C%0A%5Chline%0A5%260%7B%2C%7D361%5C%5C%0A1%260%7B%2C%7D185%5C%5C%0A-5%260%7B%2C%7D111%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l} x&P\left(X=x\right)\\ \hline 5&0{,}361\\ 1&0{,}185\\ -5&0{,}111 \end{array}"/>e</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%3D5%5Cright)%3DP%5Cleft(punainen%5C%20JA%5C%20punainen%5C%20TAI%5C%20vihre%C3%A4%5C%20JA%5C%20vihre%C3%A4%5Cright)" alt="P\left(X=5\right)=P\left(punainen\ JA\ punainen\ TAI\ vihreä\ JA\ vihreä\right)"/></div>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B5%7D%7B9%7D%5Ccdot%5Cfrac%7B4%7D%7B8%7D%2B%5Cfrac%7B3%7D%7B9%7D%5Ccdot%5Cfrac%7B2%7D%7B8%7D%3D0%7B%2C%7D3611...%5Capprox0%7B%2C%7D361" alt="=\frac{5}{9}\cdot\frac{4}{8}+\frac{3}{9}\cdot\frac{2}{8}=0{,}3611...\approx0{,}361"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%3D1%5Cright)%3DP%5Cleft(punainen%5C%20JA%5C%20Vihre%C3%A4%5Cright)%3D%5Cfrac%7B5%7D%7B9%7D%5Ccdot%5Cfrac%7B1%7D%7B3%7D%3D0%7B%2C%7D1851...%5Capprox0%7B%2C%7D185" alt="P\left(X=1\right)=P\left(punainen\ JA\ Vihreä\right)=\frac{5}{9}\cdot\frac{1}{3}=0{,}1851...\approx0{,}185"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%3D-5%5Cright)%3D%5Cfrac%7B1%7D%7B9%7D%3D0%7B%2C%7D1111...%5Capprox0%7B%2C%7D111" alt="P\left(X=-5\right)=\frac{1}{9}=0{,}1111...\approx0{,}111"/></div>
<img src="https://math-demo.abitti.fi/math.svg?latex=E%5Cleft(X%5Cright)%3D5%5Ccdot0%7B%2C%7D361%2B1%5Ccdot0%7B%2C%7D185%2B-5%5Ccdot0%7B%2C%7D111%3D1%7B%2C%7D435" alt="E\left(X\right)=5\cdot0{,}361+1\cdot0{,}185+-5\cdot0{,}111=1{,}435"/><br/>
<div><br/>
<div>429</div>
<div>430</div>
<div>433</div>
<div>435</div>
</div>
</div>
</div>
2019-10-29T09:04:17+02:00
Teksti
https://peda.net/id/dbfd1cacf6e
2019-11-14T10:56:02+02:00
<div>Kertymäfunktio</div>
<div>Satunnaismuuttujan X kertymäfunktio on </div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=F%5Cleft(x%5Cright)%3DP%5Cleft(X%5Cle%20x%5Cright)%7B%2C%7D%5C%20x%5Cin%5Cmathbb%7BR%7D" alt="F\left(x\right)=P\left(X\le x\right){,}\ x\in\mathbb{R}"/></div>
<div>Kertymäfunktio arvo kohdassa x on kohdan x jasit pienempien satunnaismuuttujan arvojen todennäköisyyksien summa</div>
<div> </div>
<div>414</div>
<div>a)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=F%5Cleft(x%5Cright)%3D%5Cbegin%7Bcases%7D%0A0%7B%2C%7D%26kun%5C%20x%3C1%5C%5C%0A0%7B%2C%7D15%7B%2C%7D%26kun%5C%201%5Cle%20x%5C%5C%0A0%7B%2C%7D25%7B%2C%7D%26kun%5C%202%5Cle%20x%5Cle3%5C%5C%0A0%7B%2C%7D5%7B%2C%7D%26kun%5C%203%5Cle%20x%5Cle4%5C%5C%0A0%7B%2C%7D8%7B%2C%7D%26kun%5C%204%5Cle%20x%5Cle5%5C%5C%0A1%7B%2C%7D0%7B%2C%7D%26kun%5C%20x%5Cge5%0A%5Cend%7Bcases%7D" alt="F\left(x\right)=\begin{cases} 0{,}&kun\ x<1\\ 0{,}15{,}&kun\ 1\le x\\ 0{,}25{,}&kun\ 2\le x\le3\\ 0{,}5{,}&kun\ 3\le x\le4\\ 0{,}8{,}&kun\ 4\le x\le5\\ 1{,}0{,}&kun\ x\ge5 \end{cases}"/></div>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%5Cle3%5Cright)%3DF%5Cleft(3%5Cright)%3D0%7B%2C%7D5" alt="P\left(X\le3\right)=F\left(3\right)=0{,}5"/></div>
<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(1%3CX%5Cle4%5Cright)%3DF%5Cleft(4%5Cright)-F%5Cleft(1%5Cright)%3D0%7B%2C%7D8-0%7B%2C%7D15%3D0%7B%2C%7D65" alt="P\left(1<X\le4\right)=F\left(4\right)-F\left(1\right)=0{,}8-0{,}15=0{,}65"/><br/>
<div>c)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=F%5Cleft(2%5Cright)%3DP%5Cleft(X%5Cle2%5Cright)%3D0%7B%2C%7D25" alt="F\left(2\right)=P\left(X\le2\right)=0{,}25"/></div>
<br/>
401<br/>
a) Pistetodennäköisyyksien summa on yksi, joten<br/>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%3D2%5Cright)%3D1-%5Cleft(%5Cfrac%7B1%7D%7B16%7D%2B%5Cfrac%7B3%7D%7B8%7D%2B%5Cfrac%7B1%7D%7B4%7D%5Cright)%3D%5Cfrac%7B5%7D%7B16%7D" alt="P\left(X=2\right)=1-\left(\frac{1}{16}+\frac{3}{8}+\frac{1}{4}\right)=\frac{5}{16}"/></div>
<span>b) Koska e < 3 ja π > 3, saadaan erillisten tapahtumien yhteenlaskusäännön mukaan </span><br/>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%5Cle3%5Cright)%3DP%5Cleft(X%3D1%5Cright)%2BP%5Cleft(X%3D2%5Cright)%2BP%5Cleft(X%3De%5Cright)%3D%5Cfrac%7B1%7D%7B16%7D%2B%5Cfrac%7B5%7D%7B16%7D%2B%5Cfrac%7B3%7D%7B8%7D%3D%5Cfrac%7B3%7D%7B4%7D" alt="P\left(X\le3\right)=P\left(X=1\right)+P\left(X=2\right)+P\left(X=e\right)=\frac{1}{16}+\frac{5}{16}+\frac{3}{8}=\frac{3}{4}"/></div>
c)<br/>
<a href="https://peda.net/p/kirin_porsti/ma/ma10p/teht%C3%A4v%C3%A4t/nimet%C3%B6n-dbfd/401-c-png#top" title="401 c.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma10p/teht%C3%A4v%C3%A4t/nimet%C3%B6n-dbfd/401-c-png:file/photo/11d64e93c024266d6cbcfb47073dc327ed03a209/401%20c.PNG" alt="" title="401 c.PNG" class="inline" loading="lazy"/></a><br/>
<br/>
<div>406</div>
<div>a)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=Q%3D1-%5Cleft(%5Cfrac%7B1%7D%7B6%7D%2B%5Cfrac%7B1%7D%7B5%7D%2B%5Cfrac%7B1%7D%7B3%7D%5Cright)%3D1-%5Cleft(%5Cfrac%7B5%7D%7B30%7D%2B%5Cfrac%7B6%7D%7B30%7D%2B%5Cfrac%7B10%7D%7B30%7D%5Cright)%3D1-%5Cfrac%7B21%7D%7B30%7D%3D%5Cfrac%7B9%7D%7B30%7D%3D%5Cfrac%7B3%7D%7B10%7D" alt="Q=1-\left(\frac{1}{6}+\frac{1}{5}+\frac{1}{3}\right)=1-\left(\frac{5}{30}+\frac{6}{30}+\frac{10}{30}\right)=1-\frac{21}{30}=\frac{9}{30}=\frac{3}{10}"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(2%7B%2C%7D%5Cfrac%7B3%7D%7B10%7D%5Cright)" alt="\left(2{,}\frac{3}{10}\right)"/></div>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0AX%26P%5Cleft(X%3Dx%5Cright)%5C%5C%0A%5Chline%0A-3%26%5Cfrac%7B1%7D%7B6%7D%5C%5C%0A2%26%5Cfrac%7B3%7D%7B10%7D%5C%5C%0A4%26%5Cfrac%7B1%7D%7B5%7D%5C%5C%0A6%26%5Cfrac%7B1%7D%7B3%7D%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l} X&P\left(X=x\right)\\ \hline -3&\frac{1}{6}\\ 2&\frac{3}{10}\\ 4&\frac{1}{5}\\ 6&\frac{1}{3} \end{array}"/></div>
c) P(0≤X≤6)
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P(0%5Cle%20X%5Cle6)%3DP%5Cleft(X%3D2%5Cright)%2BP%5Cleft(X%3D4%5Cright)%3D%5Cfrac%7B3%7D%7B10%7D%2B%5Cfrac%7B1%7D%7B5%7D%3D%5Cfrac%7B9%7D%7B30%7D%2B%5Cfrac%7B6%7D%7B30%7D%3D%5Cfrac%7B15%7D%7B30%7D%3D%5Cfrac%7B1%7D%7B2%7D" alt="P(0\le X\le6)=P\left(X=2\right)+P\left(X=4\right)=\frac{3}{10}+\frac{1}{5}=\frac{9}{30}+\frac{6}{30}=\frac{15}{30}=\frac{1}{2}"/></div>
<div><br/>
<div>409</div>
<div>412</div>
<div>415</div>
</div>
2019-10-25T08:39:48+03:00
Teksti
https://peda.net/id/1bb21df0f61
2019-10-24T08:28:28+03:00
<span>351</span>
<div>2 Ruskeaa, 6 mustaa ja 8 sisnistä</div>
<div>
<div>Kuoria on yhteensä 16 kpl</div>
<div> </div>
<div>Tapa 1:</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(samanv%C3%A4riset%5Cright)%3DP%5Cleft(%5Cmathrm%7B2%5C%20ruskeaa%5C%20TAI%5C%202%5C%20mustaa%5C%20TAI%5C%202%5C%20%5Csin%20ist%C3%A4%7D%5Cright)" alt="P\left(samanväriset\right)=P\left(\mathrm{2\ ruskeaa\ TAI\ 2\ mustaa\ TAI\ 2\ \sin istä}\right)"/></div>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3DP%5Cmathrm%7B%5Cleft(2%5C%20ruskeaa%5Cright)%2BP%5Cleft(2%5C%20mustaa%5Cright)%2BP%5Cleft(2%5C%20%5Csin%20ist%C3%A4%5Cright)%7D" alt="=P\mathrm{\left(2\ ruskeaa\right)+P\left(2\ mustaa\right)+P\left(2\ \sin istä\right)}"/><br/>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B2%7D%7B16%7D%5Ccdot%5Cfrac%7B1%7D%7B15%7D%2B%5Cfrac%7B6%7D%7B16%7D%5Ccdot%5Cfrac%7B5%7D%7B15%7D%2B%5Cfrac%7B8%7D%7B16%7D%5Ccdot%5Cfrac%7B7%7D%7B15%7D" alt="=\frac{2}{16}\cdot\frac{1}{15}+\frac{6}{16}\cdot\frac{5}{15}+\frac{8}{16}\cdot\frac{7}{15}"/></div>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B2%7D%7B240%7D%2B%5Cfrac%7B30%7D%7B240%7D%2B%5Cfrac%7B56%7D%7B240%7D" alt="=\frac{2}{240}+\frac{30}{240}+\frac{56}{240}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B88%7D%7B240%7D" alt="=\frac{88}{240}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B11%7D%7B30%7D" alt="=\frac{11}{30}"/></div>
<div>Tapa 2:</div>
<div>16 kuoren joukosta voidaan valita 2 kuorta</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B16%7D%7B2%7D%3D120" alt="\binom{16}{2}=120"/>eri tavalla</div>
<div>Kaksi ruskeaa kuorta voidaan valita yhdellä tavalla</div>
<div>Kaksi mustaa kurota voidaan valita </div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B6%7D%7B2%7D%3D15" alt="\binom{6}{2}=15"/>eri tavalla ja </div>
<div>kaksi sinistä kuorta voidaan valita </div>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B8%7D%7B2%7D%3D28" alt="\binom{8}{2}=28"/><span>eri tavalla.</span>
<div>Yhteensä samanväriset kuoret voidaan valita 1+15+28=44 eri tavalla</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(samanv%C3%A4riset%5Cright)%3D%5Cfrac%7B44%7D%7B120%7D%3D%5Cfrac%7B11%7D%7B30%7D" alt="P\left(samanväriset\right)=\frac{44}{120}=\frac{11}{30}"/></div>
<div> </div>
<div>342</div>
<div>a)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(valitaan%5C%20puheenjohtaja%5C%20JA%5C%20sihteeri%5C%20JA%5C%20taloudenhoitaja%5Cright)%3DP%5Cleft(%5Csim%5Cright)" alt="P\left(valitaan\ puheenjohtaja\ JA\ sihteeri\ JA\ taloudenhoitaja\right)=P\left(\sim\right)"/></div>
<br/>
<span>Puheenjohtajaa voidaan valita</span><br/>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B13%7D%7B1%7D%3D13" alt="\binom{13}{1}=13"/>eri tavalla</div>
<div>Sihteeriä voidaan valita</div>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B12%7D%7B1%7D%3D12" alt="\binom{12}{1}=12"/><span>eri tavalla</span>
<div>Taloudenhoitajaa voidaan valita <br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B11%7D%7B1%7D%3D11" alt="\binom{11}{1}=11"/>eri tavalla</div>
<div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%3DP%5Cleft(%5Csim%5Cright)%3D13%5Ccdot12%5Ccdot11%3D1716" alt="=P\left(\sim\right)=13\cdot12\cdot11=1716"/></div>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B3%7D%7B1%7D%2B%5Cbinom%7B2%7D%7B1%7D%2B%5Cbinom%7B1%7D%7B1%7D%3D3%2B2%2B1%3D6" alt="\binom{3}{1}+\binom{2}{1}+\binom{1}{1}=3+2+1=6"/><br/>
c)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B13%7D%7B3%7D%3D286" alt="\binom{13}{3}=286"/></div>
<div><br/>
<div>344</div>
<div>Tapoja on yhteensä</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccdot2%5Ccdot2%5Ccdot2%3D2%5E4%3D16" alt="2\cdot2\cdot2\cdot2=2^4=16"/></div>
Koska yksi on ainakin päällä, tästä vähenettävä yhden ''suljettu'' vaihto.</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=16-1%3D15" alt="16-1=15"/></div>
<div> </div>
<div>
<div>345</div>
<div>Vakioveikkauksessa valitaan 13 tulosta 3 vaihtoehtosta, valinantapoja on tuloperiaatteen mukaan</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=3%5E%7B13%7D%3D1%5C%20594%5C%20323" alt="3^{13}=1\ 594\ 323"/>, eli lähes 1,6 miljoonaa</div>
Lotossa valitaan 7 oikeita numeroa 40 vaihtoehtosta</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B40%7D%7B7%7D%3D18%5C%20643%5C%20560" alt="\binom{40}{7}=18\ 643\ 560"/>, eli lähes 19 miljoonaa</div>
<div>Joten vakioveikkauksessa päävoitto on todennäköisempi.</div>
<div> </div>
<div>
<div>346</div>
</div>
<div>Lasten lotossa voidaan valita 3 numeroa <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B10%7D%7B3%7D%3D120" alt="\binom{10}{3}=120"/> tavalla.</div>
<div>Jos halutaan saada 0 oikeata, valitaan 7 väärästä numerosta 3, eli</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B7%7D%7B3%7D%3D35%7B%2C%7D%5C%20todenn%C3%A4k%C3%B6isyys%5C%20on%5C%20%5Cfrac%7B24%7D%7B120%7D%3D%5Cfrac%7B7%7D%7B24%7D%3D0%7B%2C%7D291...%5Capprox0%7B%2C%7D30" alt="\binom{7}{3}=35{,}\ todennäköisyys\ on\ \frac{24}{120}=\frac{7}{24}=0{,}291...\approx0{,}30"/></div>
<br/>
Jos halutaan saada 1 oikean, valitaan 3 oikeasta 1 JA 7 väärästä 2, eli</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B3%7D%7B1%7D%5Ccdot%5Cbinom%7B7%7D%7B2%7D%3D3%5Ccdot42%3D126%7B%2C%7D%5C%20todenn%C3%A4k%C3%B6isyys%5C%20on%5C%20%5Cfrac%7B126%7D%7B120%7D%3D%5Cfrac%7B7%7D%7B40%7D%3D0%7B%2C%7D175%5Capprox0%7B%2C%7D18" alt="\binom{3}{1}\cdot\binom{7}{2}=3\cdot42=126{,}\ todennäköisyys\ on\ \frac{126}{120}=\frac{7}{40}=0{,}175\approx0{,}18"/></div>
<div> </div>
<div>Jos halutaan saada 2 oikeata, valitaan 3 oikeasta 2 JA 7 väärästä 1, eli</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B3%7D%7B2%7D%5Ccdot%5Cbinom%7B7%7D%7B1%7D%3D42%7B%2C%7D%5C%20todenn%C3%A4k%C3%B6isyys%5C%20on%5C%20%5Cfrac%7B42%7D%7B720%7D%3D%5Cfrac%7B7%7D%7B120%7D%3D0%7B%2C%7D583...%5Capprox0%7B%2C%7D58" alt="\binom{3}{2}\cdot\binom{7}{1}=42{,}\ todennäköisyys\ on\ \frac{42}{720}=\frac{7}{120}=0{,}583...\approx0{,}58"/></div>
<div> </div>
<div>Jos halutaa saada kaikki oikein(3 oikein), valitaan 3 oikesta 3</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B3%7D%7B3%7D%3D" alt="\binom{3}{3}="/></div>
<br/>
<br/>
<br/>
<div> </div>
<div>
<div> </div>
<div> </div>
<br/>
<div> </div>
<div><br/>
<div> </div>
<div>347</div>
<div>349</div>
<div>452</div>
<div>353</div>
<div>355</div>
<div>356</div>
<div><br/>
<div> </div>
</div>
</div>
</div>
2019-10-24T08:28:28+03:00
Teksti
https://peda.net/id/ebbd504cf61
2019-10-24T08:27:07+03:00
<span>334 </span>
<div>3 omenaa, 4 päärynää, 5 mandariinia</div>
<div>Hedemiä on yhteensä 12 kappaletta</div>
<div>12 hedelmän joukosta voidaan valita 2 hedelmää</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B12%7D%7B2%7D%3D66" alt="\binom{12}{2}=66"/>eri tavalla</div>
<div>a) </div>
<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(ainakin%5C%201%5C%20omena%5Cright)%3D1-P%5Cleft(ei%5C%20yht%C3%A4%C3%A4n%5C%20omenaa%5Cright)%3D1-%5Cfrac%7B%5Cbinom%7B9%7D%7B2%7D%7D%7B66%7D%3D1-%5Cfrac%7B36%7D%7B66%7D%3D%5Cfrac%7B5%7D%7B11%7D%5Capprox0%7B%2C%7D45" alt="P\left(ainakin\ 1\ omena\right)=1-P\left(ei\ yhtään\ omenaa\right)=1-\frac{\binom{9}{2}}{66}=1-\frac{36}{66}=\frac{5}{11}\approx0{,}45"/><br/>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(molemmat%5C%20samaa%5C%20lajia%5Cright)%3DP%5Cleft(2%5C%20omenaa%5C%20tai%5C%202%5C%20p%C3%A4%C3%A4ryn%C3%A4%C3%A4%5C%20tai%5C%202%5C%20mandariinia%5Cright)" alt="P\left(molemmat\ samaa\ lajia\right)=P\left(2\ omenaa\ tai\ 2\ päärynää\ tai\ 2\ mandariinia\right)"/></div>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B%5Cleft(%5Cbinom%7B3%7D%7B2%7D%2B%5Cbinom%7B4%7D%7B2%7D%2B%5Cbinom%7B5%7D%7B2%7D%5Cright)%7D%7B66%7D%3D%5Cfrac%7B19%7D%7B66%7D%5Capprox0%7B%2C%7D29" alt="=\frac{\left(\binom{3}{2}+\binom{4}{2}+\binom{5}{2}\right)}{66}=\frac{19}{66}\approx0{,}29"/><br/>
<div> </div>
<div>321</div>
<div>a)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B7!%7D%7B%5Cleft(7-3%5Cright)!%7D%3D210" alt="\frac{7!}{\left(7-3\right)!}=210"/></div>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B7%7D%7B3%7D%3D35" alt="\binom{7}{3}=35"/></div>
<div> </div>
<div>322</div>
<div>32 korttia</div>
<div>1. 5 korttia</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B32%7D%7B5%7D%3D201%5C%20376" alt="\binom{32}{5}=201\ 376"/></div>
<div>2. 5 korttia</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B32-5%7D%7B5%7D%3D%5Cbinom%7B27%7D%7B5%7D%3D80%5C%20730" alt="\binom{32-5}{5}=\binom{27}{5}=80\ 730"/></div>
<div> </div>
<div>325</div>
<div>10 poikaa </div>
<div>16 tyttöä</div>
<div>a)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(poika%5C%20ja%5C%20tytt%C3%B6%5C%20pari%5Cright)%3D10%5Ccdot16%3D160" alt="P\left(poika\ ja\ tyttö\ pari\right)=10\cdot16=160"/></div>
<span>b)</span>
<div>Opiskelijoita on yhteensä 10+16=26 henkilöä</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(mik%C3%A4%5C%20tahansa%5C%20pari%5Cright)%3D%5Cbinom%7B26%7D%7B2%7D%3D325" alt="P\left(mikä\ tahansa\ pari\right)=\binom{26}{2}=325"/><br/>
<div><br/>
<div>326</div>
<div>47 eläintä</div>
<div>Vilina tunnistaa 35 eläintä</div>
<div>Ei tuunista 12 eläintä</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B47%7D%7B5%7D%3D1%5C%20533%5C%20939" alt="\binom{47}{5}=1\ 533\ 939"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B35%7D%7B5%7D%3D324632" alt="\binom{35}{5}=324632"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(Vilina%5C%20tunnistaa%5C%20kaikki%5C%20viisi%5C%20el%C3%A4int%C3%A4%5Cright)%3D%5Cfrac%7B324632%7D%7B1533939%7D%3D0%7B%2C%7D211...%5Capprox0%7B%2C%7D21" alt="P\left(Vilina\ tunnistaa\ kaikki\ viisi\ eläintä\right)=\frac{324632}{1533939}=0{,}211...\approx0{,}21"/></div>
<br/>
<div>327</div>
<div>1-40</div>
<div>a)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(6%2B1%5C%20oikein%5Cright)%3D%5Cfrac%7B%5Cbinom%7B7%7D%7B6%7D%5Ccdot%5Cbinom%7B1%7D%7B1%7D%7D%7B%5Cbinom%7B40%7D%7B7%7D%7D%3D%5Cfrac%7B7%7D%7B18643560%7D" alt="P\left(6+1\ oikein\right)=\frac{\binom{7}{6}\cdot\binom{1}{1}}{\binom{40}{7}}=\frac{7}{18643560}"/></div>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(6%5C%20oikein%5Cright)%3D%5Cfrac%7B%5Cbinom%7B7%7D%7B6%7D%5Ccdot%5Cbinom%7B40-7-1%7D%7B1%7D%7D%7B%5Cbinom%7B40%7D%7B7%7D%7D%3D%5Cfrac%7B28%7D%7B2330445%7D" alt="P\left(6\ oikein\right)=\frac{\binom{7}{6}\cdot\binom{40-7-1}{1}}{\binom{40}{7}}=\frac{28}{2330445}"/></div>
<div>c)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(4%5C%20oikein%5Cright)%3D%5Cfrac%7B%5Cbinom%7B7%7D%7B4%7D%5Ccdot%5Cbinom%7B40-7-1%7D%7B3%7D%7D%7B%5Cbinom%7B40%7D%7B7%7D%7D%3D%5Cfrac%7B4340%7D%7B466089%7D" alt="P\left(4\ oikein\right)=\frac{\binom{7}{4}\cdot\binom{40-7-1}{3}}{\binom{40}{7}}=\frac{4340}{466089}"/></div>
<div> </div>
<div>329</div>
<div>5 1.opiskeljiaa</div>
<div>7 2.opiskelijaa</div>
<div>Koska delegaatioita voidaan muodosta vain kahdella tavalla:</div>
<div>2 1.opiskelijaa + 1 2.opiskelija</div>
<div>tai </div>
<div>2 2.opiskelijaa + 1 1.opiskelija</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(2%5C%201.opiskelijaa%2B1%5C%202.opiskelija%5Cright)%3D%5Cbinom%7B5%7D%7B2%7D%5Ccdot%5Cbinom%7B7%7D%7B1%7D%3D70" alt="P\left(2\ 1.opiskelijaa+1\ 2.opiskelija\right)=\binom{5}{2}\cdot\binom{7}{1}=70"/></div>
<div>ja</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(2%5C%202.opiskelijaa%2B1%5C%201.opiskelija%5Cright)%3D%5Cbinom%7B7%7D%7B2%7D%5Ccdot%5Cbinom%7B5%7D%7B1%7D%3D105" alt="P\left(2\ 2.opiskelijaa+1\ 1.opiskelija\right)=\binom{7}{2}\cdot\binom{5}{1}=105"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=105%2B70%3D175" alt="105+70=175"/></div>
V: Delegaatioita voidaan mudosta 175 eri tavalla.<br/>
<div> </div>
<div>332</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B20%7D%7B4%7D%3D%5Cfrac%7B20!%7D%7B4!%5Cleft(20-4%5Cright)!%7D%3D%5Cfrac%7B20!%7D%7B4!%5Ccdot16!%7D%3D%5Cfrac%7B20%5Ccdot19%5Ccdot18%5Ccdot17%7D%7B4%5Ccdot3%5Ccdot2%5Ccdot1%7D%3D%5Cfrac%7B5%5Ccdot19%5Ccdot3%5Ccdot17%7D%7B1%7D%3D19%5Ccdot17%5Ccdot15" alt="\binom{20}{4}=\frac{20!}{4!\left(20-4\right)!}=\frac{20!}{4!\cdot16!}=\frac{20\cdot19\cdot18\cdot17}{4\cdot3\cdot2\cdot1}=\frac{5\cdot19\cdot3\cdot17}{1}=19\cdot17\cdot15"/></div>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B20%7D%7B12%7D%3D%5Cfrac%7B20!%7D%7B12!%5Cleft(20-12%5Cright)!%7D%3D%5Cfrac%7B20!%7D%7B12!%5Ccdot8!%7D%3D%5Cfrac%7B20%5Ccdot19%5Ccdot18%5Ccdot17%5Ccdot16%5Ccdot15%5Ccdot14%5Ccdot13%7D%7B8%5Ccdot7%5Ccdot6%5Ccdot5%5Ccdot4%5Ccdot3%5Ccdot2%5Ccdot1%7D%3D%5Cfrac%7B19%5Ccdot17%5Ccdot15%5Ccdot14%5Ccdot13%7D%7B7%7D" alt="\binom{20}{12}=\frac{20!}{12!\left(20-12\right)!}=\frac{20!}{12!\cdot8!}=\frac{20\cdot19\cdot18\cdot17\cdot16\cdot15\cdot14\cdot13}{8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}=\frac{19\cdot17\cdot15\cdot14\cdot13}{7}"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B20%7D%7B19%7D%3D%5Cfrac%7B20!%7D%7B19!%5Cleft(20-19%5Cright)!%7D%3D" alt="\binom{20}{19}=\frac{20!}{19!\left(20-19\right)!}="/><br/>
<div>333</div>
<div>337 </div>
<div>338</div>
</div>
</div>
2019-10-24T08:27:07+03:00
Teksti
https://peda.net/id/6d06bd20eb1
2019-10-10T08:25:52+03:00
<div>302</div>
<div>3 housua</div>
<div>2 pitkähihaista paitaa</div>
<div>1 musta t-paita</div>
<div>1 vaaleapunainen t-paita</div>
<div>2 valkoista t-paitaa</div>
<div>a)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2%7D%7B6%7D%5Ccdot%5Cfrac%7B1%7D%7B6%7D%3D%5Cfrac%7B1%7D%7B18%7D" alt="\frac{2}{6}\cdot\frac{1}{6}=\frac{1}{18}"/></div>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7B3%7D%5Ccdot%5Cfrac%7B2%7D%7B6%7D%3D%5Cfrac%7B1%7D%7B9%7D" alt="\frac{1}{3}\cdot\frac{2}{6}=\frac{1}{9}"/></div>
<span>c)</span>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=1%5Ccdot%5Cfrac%7B2%7D%7B6%7D%3D%5Cfrac%7B2%7D%7B6%7D%3D%5Cfrac%7B1%7D%7B3%7D" alt="1\cdot\frac{2}{6}=\frac{2}{6}=\frac{1}{3}"/><br/>
<br/>
303
<div>a) Nlejästä henkilöstä valitaan juoksijat 4-osuuksiseen viestiin, </div>
<div>Erilaisia juoksujärjestyksiä eli permutaatioita on </div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=4!%3D4%5Ccdot3%5Ccdot2%5Ccdot1%3D24" alt="4!=4\cdot3\cdot2\cdot1=24"/></div>
<div>b) 7 henkilöstä valitaan juoksijat 5-osuuksiseen viestiin. </div>
<div>Erilaisia juoksujärjestyksiä on </div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=7%5Ccdot6%5Ccdot5%5Ccdot4%5Ccdot3%3D2520" alt="7\cdot6\cdot5\cdot4\cdot3=2520"/></div>
Tapa 2:
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(n%5Cright)_k%3D%5Cfrac%7Bn!%7D%7B%5Cleft(n-k%5Cright)!%7D%3Dn%5Ccdot%5Cleft(n-1%5Cright)%5Ccdot...%5Ccdot%5Cleft(n-k%2B1%5Cright)" alt="\left(n\right)_k=\frac{n!}{\left(n-k\right)!}=n\cdot\left(n-1\right)\cdot...\cdot\left(n-k+1\right)"/></div>
<div>7-alkioisen joukon 5-permutaatiota on </div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(7%5Cright)_5%3D%5Cfrac%7B7!%7D%7B%5Cleft(7-5%5Cright)!%7D%3D%5Cfrac%7B7!%7D%7B2!%7D%3D%5Cfrac%7B7%5Ccdot6%5Ccdot5%5Ccdot4%5Ccdot3%5Ccdot2%5Ccdot1%7D%7B2%5Ccdot1%7D%3D7%5Ccdot6%5Ccdot5%5Ccdot4%5Ccdot3%3D2520" alt="\left(7\right)_5=\frac{7!}{\left(7-5\right)!}=\frac{7!}{2!}=\frac{7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}{2\cdot1}=7\cdot6\cdot5\cdot4\cdot3=2520"/>kappaletta.</div>
<div> </div>
<div>304</div>
<div>a)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=1.%3D9" alt="1.=9"/></div>
<img src="https://math-demo.abitti.fi/math.svg?latex=2.%3D10" alt="2.=10"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=3.%3D10" alt="3.=10"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=4.%3D2" alt="4.=2"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=9%5Ccdot10%5Ccdot10%5Ccdot2%3D1800" alt="9\cdot10\cdot10\cdot2=1800"/></div>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1800%7D%7B10000-1000%7D%3D%5Cfrac%7B1%7D%7B5%7D" alt="\frac{1800}{10000-1000}=\frac{1}{5}"/></div>
<div><br/>
<div>306</div>
<div>a)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B10%5Ccdot9%5Ccdot8%5Ccdot7%5Ccdot6%5Ccdot5%5Ccdot4%5Ccdot3%5Ccdot2%5Ccdot1%7D%7B7%5Ccdot6%5Ccdot5%5Ccdot4%5Ccdot3%5Ccdot2%5Ccdot1%7D%3D10%5Ccdot9%5Ccdot8%3D720" alt="\frac{10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}{7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}=10\cdot9\cdot8=720"/></div>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(14%5Cright)_4%3D%5Cfrac%7B14!%7D%7B%5Cleft(14-4%5Cright)!%7D%3D%3D%5Cfrac%7B14!%7D%7B10!%7D" alt="\left(14\right)_4=\frac{14!}{\left(14-4\right)!}==\frac{14!}{10!}"/></div>
<br/>
<div>309</div>
<div>5 tyttöä</div>
<div>4 poikaa</div>
<div>a)</div>
<div>Aloittaja on tyttö=joka toinen on poika</div>
</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=5%5Ccdot4%5Ccdot4%5Ccdot3%5Ccdot3%3D720" alt="5\cdot4\cdot4\cdot3\cdot3=720"/></div>
<div>Aloittajat on poika=joka toinen on tyttö</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=4%5Ccdot5%5Ccdot3%5Ccdot4%5Ccdot2%3D480" alt="4\cdot5\cdot3\cdot4\cdot2=480"/></div>
<span>720+480=1200</span>
<div>b)</div>
<div>Lapsia on yhteensä 9, joten niitä 9-alkioisen joukon 5-permutaatiota on </div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(9%5Cright)_5%3D15120" alt="\left(9\right)_5=15120"/></div>
<div>Koska jonoja on mahdollista muodosta 1200 kpl, sen todennäköisyys saadaan</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1200%7D%7B15120%7D%3D%5Cfrac%7B5%7D%7B63%7D" alt="\frac{1200}{15120}=\frac{5}{63}"/></div>
<div>
<div> </div>
<div>310</div>
<div>Oikean pääsykoodin vaihtoehtoja on</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=1.%5C%202" alt="1.\ 2"/></div>
<img src="https://math-demo.abitti.fi/math.svg?latex=2.%5C%2010" alt="2.\ 10"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=3.%5C%2010" alt="3.\ 10"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=4.%5C%202" alt="4.\ 2"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccdot10%5Ccdot10%5Ccdot2%3D400" alt="2\cdot10\cdot10\cdot2=400"/></div>
<div>Milla aikoo kokeilla eri numerosarjoja, joten hän valitsee kokeiltaviksi kolme eri numerosarjaa. Todennäköisyys, että oikea numerosarja on jokin näistä kolmesta, on </div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B3%7D%7B400%7D" alt="=\frac{3}{400}"/></div>
<div>
<div> </div>
<div>311</div>
<div>313</div>
<div>314</div>
<div>
<div>
<div> </div>
</div>
</div>
</div>
2019-10-10T08:25:52+03:00
Teksti
https://peda.net/id/4f464c2eeb1
2019-10-10T08:25:02+03:00
<div>265</div>
<div>Karjalanpiirkoita 12</div>
<div>Pikkupullia 9</div>
<div>7 molempia</div>
<div>5 ei kumpikaan</div>
<div>a)</div>
<div>12+2+5=19</div>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(leipoi%5C%20karjalanpiirakoita%5C%20mutta%5C%20ei%5C%20pikkupullia%5Cright)%3D%5Cfrac%7B5%7D%7B19%7D%5Capprox0%7B%2C%7D263...%5Capprox0%7B%2C%7D26" alt="P\left(leipoi\ karjalanpiirakoita\ mutta\ ei\ pikkupullia\right)=\frac{5}{19}\approx0{,}263...\approx0{,}26"/></div>
<br/>
<span>266 </span>
<div>Asukkaita 20% on kauniita, 15% rohkeita ja 5% kauniita ja rohkeita.</div>
<div>a)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(rohkea%5C%20mutta%5C%20ei%5C%20kaunis%5Cright)%3D10%5C%25%3D0%7B%2C%7D1" alt="P\left(rohkea\ mutta\ ei\ kaunis\right)=10\%=0{,}1"/></div>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(joko%5C%20kaunis%5C%20tai%5C%20rohkea%7B%2C%7D%5C%20mutta%5C%20ei%5C%20molempia%5Cright)%3D15%5C%25%2B10%5C%25%3D25%5C%25%3D0%7B%2C%7D25" alt="P\left(joko\ kaunis\ tai\ rohkea{,}\ mutta\ ei\ molempia\right)=15\%+10\%=25\%=0{,}25"/></div>
<span>c) </span>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(kaunis%5C%20tai%5C%20rohkea%5Cright)%3D15%5C%25%2B5%5C%25%2B10%5C%25%3D30%5C%25%3D0%7B%2C%7D3" alt="P\left(kaunis\ tai\ rohkea\right)=15\%+5\%+10\%=30\%=0{,}3"/></div>
<div>d)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(ei%5C%20kaunis%5C%20eik%C3%A4%5C%20rohkea%5Cright)%3D1-P%5Cleft(kaunis%5C%20tai%5C%20rohkea%5Cright)%3D1-0%7B%2C%7D3%3D0%7B%2C%7D7" alt="P\left(ei\ kaunis\ eikä\ rohkea\right)=1-P\left(kaunis\ tai\ rohkea\right)=1-0{,}3=0{,}7"/></div>
<div>267</div>
<div>P(A)=Uudempi toimii todennäikösyydellä 0,95</div>
<div>P(B)=Vanhempi toimii todennäköisyydellä 0,75</div>
<div>a) </div>
<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(Uudempi%5C%20toimii%5C%20ja%5C%20vanhemi%5C%20toimii%5Cright)%3D0%7B%2C%7D95%5Ccdot0%7B%2C%7D75%3D0%7B%2C%7D7125" alt="P\left(Uudempi\ toimii\ ja\ vanhemi\ toimii\right)=0{,}95\cdot0{,}75=0{,}7125"/>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(vain%5C%20toinen%5C%20toimii%5Cright)%3DP%5Cleft(A%5C%20ja%5C%20%5Coverline%7BB%7D%5Cright)%5C%20tai%5C%20P%5Cleft(%5Coverline%7BA%7D%5C%20ja%5C%20B%5Cright)%3D0%7B%2C%7D95%5Ccdot0%7B%2C%7D25%2B0%7B%2C%7D05%5Ccdot0%7B%2C%7D75%3D0%7B%2C%7D275" alt="P\left(vain\ toinen\ toimii\right)=P\left(A\ ja\ \overline{B}\right)\ tai\ P\left(\overline{A}\ ja\ B\right)=0{,}95\cdot0{,}25+0{,}05\cdot0{,}75=0{,}275"/><br/>
<div>c) </div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(kumpikaan%5C%20ei%5C%20her%C3%A4t%C3%A4%5Cright)%3DP%5Cleft(%5Coverline%7BA%7D%5Cright)%5C%20ja%5C%20P%5Cleft(%5Coverline%7BB%7D%5Cright)%3D0%7B%2C%7D25%5Ccdot0%7B%2C%7D05%3D0%7B%2C%7D0125" alt="P\left(kumpikaan\ ei\ herätä\right)=P\left(\overline{A}\right)\ ja\ P\left(\overline{B}\right)=0{,}25\cdot0{,}05=0{,}0125"/></div>
d) </div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(ainakin%5C%20yksi%5C%20toimii%5Cright)%3DP%5Cleft(%5Coverline%7Bkumpikaan%5C%20ei%5C%20toimii%7D%5Cright)%3D1-0%7B%2C%7D0125%3D0%7B%2C%7D9875" alt="P\left(ainakin\ yksi\ toimii\right)=P\left(\overline{kumpikaan\ ei\ toimii}\right)=1-0{,}0125=0{,}9875"/></div>
<div><br/>
<div>268</div>
<div>S=sinililja itää P(S)=90%=0,9</div>
<div>T=tulppaani itää, P(T)=95%=0,95<br/>
a) </div>
<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(S%5C%20ja%5C%20T%5Cright)%3DP%5Cleft(S%5Cright)%5Ccdot%20P%5Cleft(T%5Cright)%3D0%7B%2C%7D9%5Ccdot0%7B%2C%7D95%3D0%7B%2C%7D855" alt="P\left(S\ ja\ T\right)=P\left(S\right)\cdot P\left(T\right)=0{,}9\cdot0{,}95=0{,}855"/><br/>
<div>b) </div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(S%5C%20ja%5C%20%5Coverline%7BT%7D%5Cright)%3DP%5Cleft(S%5Cright)%5Ccdot%20P%5Cleft(%5Coverline%7BT%7D%5Cright)%3D0%7B%2C%7D9%5Ccdot%5Cleft(1-0%7B%2C%7D95%5Cright)%3D0%7B%2C%7D045" alt="P\left(S\ ja\ \overline{T}\right)=P\left(S\right)\cdot P\left(\overline{T}\right)=0{,}9\cdot\left(1-0{,}95\right)=0{,}045"/></div>
c)<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(t%C3%A4sm%C3%A4lleen%5C%20yksi%5C%20it%C3%A4%C3%A4%5Cright)%3DP%5Cleft(S%5C%20ja%5C%20%5Coverline%7BT%7D%5C%20tai%5C%20%5Coverline%7BS%7D%5C%20ja%5C%20T%5Cright)%3DP%5Cleft(S%5C%20ja%5C%20%5Coverline%7BT%7D%5Cright)%2BP%5Cleft(%5Coverline%7BS%7D%2BT%5Cright)%3DP%5Cleft(S%5Cright)%5Ccdot%20P%5Cleft(%5Coverline%7BT%7D%5Cright)%2BP%5Cleft(%5Coverline%7BS%7D%5Cright)%5Ccdot%20P%5Cleft(T%5Cright)%3D0%7B%2C%7D9%5Ccdot0%7B%2C%7D05%2B0%7B%2C%7D1%5Ccdot0%7B%2C%7D95%3D0%7B%2C%7D14" alt="P\left(täsmälleen\ yksi\ itää\right)=P\left(S\ ja\ \overline{T}\ tai\ \overline{S}\ ja\ T\right)=P\left(S\ ja\ \overline{T}\right)+P\left(\overline{S}+T\right)=P\left(S\right)\cdot P\left(\overline{T}\right)+P\left(\overline{S}\right)\cdot P\left(T\right)=0{,}9\cdot0{,}05+0{,}1\cdot0{,}95=0{,}14"/><br/>
<div>d) </div>
<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(ainakin%5C%20toinen%5C%20it%C3%A4%C3%A4%5Cright)%3D1-P%5Cleft(kumpikaan%5C%20ei%5C%20id%C3%A4%5Cright)%3D1-P%5Cleft(%5Coverline%7BS%7D%5C%20ja%5C%20%5Coverline%7BT%7D%5Cright)%3D1-0%7B%2C%7D2%5Ccdot0%7B%2C%7D05%3D0%7B%2C%7D995" alt="P\left(ainakin\ toinen\ itää\right)=1-P\left(kumpikaan\ ei\ idä\right)=1-P\left(\overline{S}\ ja\ \overline{T}\right)=1-0{,}2\cdot0{,}05=0{,}995"/><br/>
<div> </div>
<div> </div>
<div>269</div>
<div>270</div>
<div>271</div>
<div>272</div>
<div>273</div>
<div>275</div>
<div>278</div>
<div>283</div>
</div>
<div> </div>
2019-10-10T08:25:02+03:00
Teksti
https://peda.net/id/38763df6eb1
2019-10-10T08:24:24+03:00
<div>243</div>
<div>a) </div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(A%5Cright)%3D%5Cfrac%7B1%7D%7B10%7D%5Ccdot%5Cfrac%7B1%7D%7B4%7D%3D%5Cfrac%7B1%7D%7B40%7D%3D0%7B%2C%7D025" alt="P\left(A\right)=\frac{1}{10}\cdot\frac{1}{4}=\frac{1}{40}=0{,}025"/></div>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(A%5Cright)%3D%5Cfrac%7B9%7D%7B10%7D%5Ccdot%5Cfrac%7B1%7D%7B4%7D%3D%5Cfrac%7B9%7D%7B40%7D%3D0%7B%2C%7D225" alt="P\left(A\right)=\frac{9}{10}\cdot\frac{1}{4}=\frac{9}{40}=0{,}225"/></div>
<span>c)</span>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(A%5Cright)%3D%5Cfrac%7B9%7D%7B10%7D%5Ccdot%5Cfrac%7B3%7D%7B4%7D%3D%5Cfrac%7B27%7D%7B40%7D%3D0%7B%2C%7D675" alt="P\left(A\right)=\frac{9}{10}\cdot\frac{3}{4}=\frac{27}{40}=0{,}675"/></div>
<div><br/>
<div>
<div>245</div>
<div>a) </div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(A%5Cright)%3DP%5Cleft(1.%5C%20valo%5C%20vihre%C3%A4%5C%20ja%5C%202.%5C%20valo%5C%20vihre%C3%A4%5C%20ja%5C%203.%5C%20valo%5C%20vihre%C3%A4%5Cright)" alt="P\left(A\right)=P\left(1.\ valo\ vihreä\ ja\ 2.\ valo\ vihreä\ ja\ 3.\ valo\ vihreä\right)"/></div>
<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(A%5Cright)%3D0%7B%2C%7D4%5Ccdot0%7B%2C%7D3%5Ccdot0%7B%2C%7D6%3D0%7B%2C%7D072" alt="P\left(A\right)=0{,}4\cdot0{,}3\cdot0{,}6=0{,}072"/></div>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(A%5Cright)%3DP%5Cleft(1.%5C%20valo%5C%20punainen%5C%20ja%5C%202.%5C%20valo%5C%20punainen%5C%20ja%5C%203.%5C%20valo%5C%20punainen%5Cright)" alt="P\left(A\right)=P\left(1.\ valo\ punainen\ ja\ 2.\ valo\ punainen\ ja\ 3.\ valo\ punainen\right)"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D0%7B%2C%7D6%5Ccdot0%7B%2C%7D7%5Ccdot0%7B%2C%7D4%3D0%7B%2C%7D168" alt="=0{,}6\cdot0{,}7\cdot0{,}4=0{,}168"/></div>
<div>c) A= Ainakin yhden punaisen valon</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BA%7D%5C%20%3DEi%5C%20ollenkaan%5C%20punaista%5C%20valoa%3D3%5C%20vihre%C3%A4t%C3%A4%5C%20" alt="\overline{A}\ =Ei\ ollenkaan\ punaista\ valoa=3\ vihreätä\ "/></div>
<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(A%5Cright)%3D1-P%5Cleft(%5Coverline%7BA%7D%5Cright)%3D1-0%7B%2C%7D072%3D0%7B%2C%7D928" alt="P\left(A\right)=1-P\left(\overline{A}\right)=1-0{,}072=0{,}928"/></div>
<div>d) </div>
<div> A= Ainakin yhden vihreän valon<br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BA%7D%5C%20%3DEi%5C%20ollenkaan%5C%20vihre%C3%A4t%C3%A4%5C%20valoa%3D3%5C%20punaista" alt="\overline{A}\ =Ei\ ollenkaan\ vihreätä\ valoa=3\ punaista"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(A%5Cright)%3D1-P%5Cleft(%5Coverline%7BA%7D%5Cright)%3D1-0%7B%2C%7D168%3D0%7B%2C%7D832" alt="P\left(A\right)=1-P\left(\overline{A}\right)=1-0{,}168=0{,}832"/> </div>
<div><br/>
<div>
<div>246</div>
<div>a)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(A%5Cright)%3D%5Cfrac%7B1%7D%7B9%7D%5Ccdot%5Cfrac%7B1%7D%7B8%7D%3D%5Cfrac%7B1%7D%7B72%7D" alt="P\left(A\right)=\frac{1}{9}\cdot\frac{1}{8}=\frac{1}{72}"/></div>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(A%5Cright)%3D%5Cfrac%7B5%7D%7B9%7D%5Ccdot%5Cfrac%7B4%7D%7B8%7D%3D%5Cfrac%7B5%7D%7B18%7D" alt="P\left(A\right)=\frac{5}{9}\cdot\frac{4}{8}=\frac{5}{18}"/></div>
c) A= Enintään yksi konsonantti</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BA%7D%3Dkaksi%5C%20konsonanttia%3D%5Cfrac%7B5%7D%7B18%7D" alt="\overline{A}=kaksi\ konsonanttia=\frac{5}{18}"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(A%5Cright)%3D1-P%5Cleft(%5Coverline%7BA%7D%5Cright)%3D1-%5Cfrac%7B5%7D%7B18%7D%3D%5Cfrac%7B13%7D%7B18%7D" alt="P\left(A\right)=1-P\left(\overline{A}\right)=1-\frac{5}{18}=\frac{13}{18}"/><br/>
<br/>
<div>248</div>
<div> </div>
<div>250</div>
<div>251</div>
<div>253</div>
<div>258</div>
<div>261</div>
</div>
</div>
2019-10-10T08:24:24+03:00
Teksti
https://peda.net/id/dfb8a500eb1
2019-11-04T00:49:04+02:00
<span>223</span>
<div>1800:</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B15231%7D%7B15231%2B15231%7D%3D0%7B%2C%7D4891...%5Capprox0%7B%2C%7D489%3D48%7B%2C%7D9%5C%25" alt="\frac{15231}{15231+15231}=0{,}4891...\approx0{,}489=48{,}9\%"/></div>
<div>1900:</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B41904%7D%7B41904%2B44435%7D%3D0%7B%2C%7D4853...%5Capprox0%7B%2C%7D485%3D48%7B%2C%7D5%5C%25" alt="\frac{41904}{41904+44435}=0{,}4853...\approx0{,}485=48{,}5\%"/></div>
<div>2000:</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B27492%7D%7B27492%2B29250%7D%3D0%7B%2C%7D4845...%5Capprox0%7B%2C%7D485%3D48%7B%2C%7D5%5C%25" alt="\frac{27492}{27492+29250}=0{,}4845...\approx0{,}485=48{,}5\%"/></div>
<span>Todennäköisyys on laskennut</span>
<div><br/>
<div>
<div>224</div>
<div>a)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B6%7D%7B10%7D%3D%5Cfrac%7B3%7D%7B5%7D" alt="\frac{6}{10}=\frac{3}{5}"/></div>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=y%3C%5Cfrac%7Bx%7D%7B3%7D" alt="y<\frac{x}{3}"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=A%3D%5Cfrac%7B6%5Ccdot2%7D%7B2%7D%2B2%5Ccdot4%3D14" alt="A=\frac{6\cdot2}{2}+2\cdot4=14"/></div>
<img src="https://math-demo.abitti.fi/math.svg?latex=A_%7Bkok%7D%3D2%5Ccdot10%3D20" alt="A_{kok}=2\cdot10=20"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(A%5Cright)%3D%5Cfrac%7B14%7D%7B20%7D%3D%5Cfrac%7B7%7D%7B10%7D" alt="P\left(A\right)=\frac{14}{20}=\frac{7}{10}"/></div>
<div><br/>
<div>225</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=A%3D%5Cint_%7B-2%7D%5E1%5Cleft(x%5E2%2B2x%2B2%5Cright)dx%3D6" alt="A=\int_{-2}^1\left(x^2+2x+2\right)dx=6"/>(Laskin)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=A_%7Bkok%7D%3D3%5Ccdot5%3D15" alt="A_{kok}=3\cdot5=15"/></div>
<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(A%5Cright)%3D%5Cfrac%7B6%7D%7B15%7D%3D%5Cfrac%7B2%7D%7B5%7D" alt="P\left(A\right)=\frac{6}{15}=\frac{2}{5}"/></div>
<div><br/>
<div>227</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B360-%5Cleft(70%5Ccdot3%2B20%5Ccdot3%5Cright)%7D%7B360%7D%3D%5Cfrac%7B1%7D%7B4%7D" alt="\frac{360-\left(70\cdot3+20\cdot3\right)}{360}=\frac{1}{4}"/></div>
<div>
<div> </div>
<div>
<div>228<br/>
a)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(A%5Cright)%3D%5Cfrac%7BA_%7B10%7D%7D%7BA_%7Bkok%7D%7D%3D%5Cfrac%7B%5Cpi%5Ccdot17%7B%2C%7D5%5E2%7D%7B%5Cpi%5Ccdot170%7B%2C%7D5%5E2%7D%3D0%7B%2C%7D0105...%5Capprox0%7B%2C%7D01" alt="P\left(A\right)=\frac{A_{10}}{A_{kok}}=\frac{\pi\cdot17{,}5^2}{\pi\cdot170{,}5^2}=0{,}0105...\approx0{,}01"/></div>
<span>b)</span><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=P(enint%C3%A4%C3%A4n%5C%205)" alt="P(enintään\ 5)"/><span>, joten </span><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BP%7D%5Cleft(%3E5%5Cright)" alt="\overline{P}\left(>5\right)"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff; color: #333333; font-family: 'Times New Roman'; font-size: 17px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;"--><br/>
<div>Luvut, jotka ovat suurmpi kuin 5 on 6, 7, 8, 9, 10</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BP%7D%5Cleft(A%5Cright)%3D%5Cfrac%7B%5Cpi%5Ccdot%5Cleft(4%5Ccdot17%2B17%7B%2C%7D5%5Cright)%5E2%7D%7B%5Cpi%5Ccdot170%7B%2C%7D5%5E2%7D%3D0%7B%2C%7D251...%5Capprox0%7B%2C%7D25" alt="\overline{P}\left(A\right)=\frac{\pi\cdot\left(4\cdot17+17{,}5\right)^2}{\pi\cdot170{,}5^2}=0{,}251...\approx0{,}25"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%3D1-%5Coverline%7BP%7D%3D1-0%7B%2C%7D25%3D0%7B%2C%7D75" alt="P=1-\overline{P}=1-0{,}25=0{,}75"/></div>
<div>c)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(1%7B%2C%7D2%5Cright)%3D%5Cfrac%7B%5Cpi%5Ccdot170%7B%2C%7D5%5E2-%5Cpi%5Ccdot%5Cleft(7%5Ccdot17%2B17%7B%2C%7D5%5Cright)%5E2%7D%7B%5Cpi%5Ccdot170%7B%2C%7D5%5E2%7D%5Capprox0%7B%2C%7D36" alt="P\left(1{,}2\right)=\frac{\pi\cdot170{,}5^2-\pi\cdot\left(7\cdot17+17{,}5\right)^2}{\pi\cdot170{,}5^2}\approx0{,}36"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(3%7B%2C%7D4%7B%2C%7D5%5Cright)%3D0%7B%2C%7D75-0%7B%2C%7D36%3D0%7B%2C%7D39" alt="P\left(3{,}4{,}5\right)=0{,}75-0{,}36=0{,}39"/></div>
</div>
</div>
</div>
<div>
<div>229</div>
<div>a)<br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B70700%2B688100%7D%7B70700%2B688100%2B789700%2B843700%2B43300%2B61800%7D%3D0%7B%2C%7D445...%5Capprox0%7B%2C%7D45%3D45%5C%25" alt="\frac{70700+688100}{70700+688100+789700+843700+43300+61800}=0{,}445...\approx0{,}45=45\%"/><br/>
b)<br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B688100%7D%7B70700%2B688100%2B789700%2B843700%2B43300%2B61800%7D%3D0%7B%2C%7D2195...%5Capprox0%7B%2C%7D22%3D22%5C%25" alt="\frac{688100}{70700+688100+789700+843700+43300+61800}=0{,}2195...\approx0{,}22=22\%"/><br/>
c)</div>
<div>
<div> </div>
<div>
<div>230</div>
<div>a) Epätosi. Tilaston perusteella kyseisen ampumahiihtäjä osuu tauluun paljon väitettyä todennäköisemmin.</div>
<div>b) Tosi, koska ampumahiihtäjän todennäköisyys osua on <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B344%7D%7B382%7D%3D0%7B%2C%7D90052...%5Capprox0%7B%2C%7D9" alt="\frac{344}{382}=0{,}90052...\approx0{,}9"/></div>
</div>
<div>c) Tosi, koska 1-0,9=0,1</div>
<div>d) Epätosi. Tilaston perusteella ei voida sanoa mitään varma yksittäisen laukauksen onnistumisesta.</div>
<div><br/>
<div>231 </div>
<div>a)</div>
<div>Koko tapahtuma kestää 65 minuuttia, ja itse purkkaus 5 minuttia, joten</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(saappuu%5C%20purkauksen%5C%20aikana%5Cright)%3D%5Cfrac%7B5%7D%7B65%7D%3D%5Cfrac%7B1%7D%7B13%7D%3D0%7B%2C%7D0769...%5Capprox0%7B%2C%7D08" alt="P\left(saappuu\ purkauksen\ aikana\right)=\frac{5}{65}=\frac{1}{13}=0{,}0769...\approx0{,}08"/></div>
<div>b) </div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(joutuu%5C%20odottaamaan%5C%20purkauksen%5C%20alkua%5C%20v%C3%A4hint%C3%A4%C3%A4n%5C%2015%5C%20%5Cmin%20s%5Cright)%3D%5Cfrac%7B60-15%2B5%7D%7B65%7D%3D%5Cfrac%7B50%7D%7B65%7D%3D%5Cfrac%7B10%7D%7B13%7D%3D0%7B%2C%7D7692...%5Capprox0%7B%2C%7D77" alt="P\left(joutuu\ odottaamaan\ purkauksen\ alkua\ vähintään\ 15\ \min s\right)=\frac{60-15+5}{65}=\frac{50}{65}=\frac{10}{13}=0{,}7692...\approx0{,}77"/></div>
c)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(joutuu%5C%20odottaamaan%5C%20purkauksen%5C%20alkua%5C%20v%C3%A4hint%C3%A4%C3%A4n%5C%2010%5C%20%5Cmin%20s%5Cright)%3D%5Cfrac%7B60-50%7D%7B65%7D%3D%5Cfrac%7B10%7D%7B65%7D%3D%5Cfrac%7B2%7D%7B13%7D%3D0%7B%2C%7D1538...%5Capprox0%7B%2C%7D15" alt="P\left(joutuu\ odottaamaan\ purkauksen\ alkua\ vähintään\ 10\ \min s\right)=\frac{60-50}{65}=\frac{10}{65}=\frac{2}{13}=0{,}1538...\approx0{,}15"/><br/>
<br/>
<div>233</div>
<div>a)</div>
<div>koska yhden pisteen pinta-ala on 0, kysytty todennäköisyys on 0</div>
<div>b)</div>
<div><br/>
237</div>
</div>
</div>
</div>
</div>
2019-10-10T08:21:55+03:00
Teksti
https://peda.net/id/fdbcfbf2e11
2019-11-03T23:54:37+02:00
<span>201</span>
<div>a) 2/3</div>
<div>b) 1/3</div>
<div>c) 1/3</div>
<div><br/>
<div>202</div>
<div>a) 1/4</div>
<div>b) 3/4</div>
<div>c) 1/2</div>
<div>d) 3/13</div>
<div> </div>
<div>204</div>
<div>a) P(A)=5/18</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%26%26%26%26%26%26%5C%5C%0A%5Chline%0A6%26x%26x%26x%26x%26x%26%5C%5C%0A5%26%26%26%26%26%26x%5C%5C%0A4%26%26%26%26%26%26x%5C%5C%0A3%26%26%26%26%26%26x%5C%5C%0A2%26%26%26%26%26%26x%5C%5C%0A1%26%26%26%26%26%26x%5C%5C%0A%261%262%263%264%265%266%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l} &&&&&&\\ \hline 6&x&x&x&x&x&\\ 5&&&&&&x\\ 4&&&&&&x\\ 3&&&&&&x\\ 2&&&&&&x\\ 1&&&&&&x\\ &1&2&3&4&5&6 \end{array}"/> </div>
<div>b) P(A)=11/36</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%26%26%26%26%26%26%5C%5C%0A%5Chline%0A6%26x%26x%26x%26x%26x%26x%5C%5C%0A5%26%26%26%26%26%26x%5C%5C%0A4%26%26%26%26%26%26x%5C%5C%0A3%26%26%26%26%26%26x%5C%5C%0A2%26%26%26%26%26%26x%5C%5C%0A1%26%26%26%26%26%26x%5C%5C%0A%261%262%263%264%265%266%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l} &&&&&&\\ \hline 6&x&x&x&x&x&x\\ 5&&&&&&x\\ 4&&&&&&x\\ 3&&&&&&x\\ 2&&&&&&x\\ 1&&&&&&x\\ &1&2&3&4&5&6 \end{array}"/> </div>
<div>c)P(A)=25/36</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%26%26%26%26%26%26%5C%5C%0A%5Chline%0A6%26%26%26%26%26%26%5C%5C%0A5%26x%26x%26x%26x%26x%26%5C%5C%0A4%26x%26x%26x%26x%26x%26%5C%5C%0A3%26x%26x%26x%26x%26x%26%5C%5C%0A2%26x%26x%26x%26x%26x%26%5C%5C%0A1%26x%26x%26x%26x%26x%26%5C%5C%0A%261%262%263%264%265%266%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l} &&&&&&\\ \hline 6&&&&&&\\ 5&x&x&x&x&x&\\ 4&x&x&x&x&x&\\ 3&x&x&x&x&x&\\ 2&x&x&x&x&x&\\ 1&x&x&x&x&x&\\ &1&2&3&4&5&6 \end{array}"/> </div>
<div>d) P(A)= 5/36</div>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%26%26%26%26%26%26%5C%5C%0A%5Chline%0A6%26%26%26%26%26%26%5C%5C%0A5%26x%26%26%26%26%26%5C%5C%0A4%26%26x%26%26%26%26%5C%5C%0A3%26%26%26x%26%26%26%5C%5C%0A2%26%26%26%26x%26%26%5C%5C%0A1%26%26%26%26%26x%26%5C%5C%0A%261%262%263%264%265%266%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l} &&&&&&\\ \hline 6&&&&&&\\ 5&x&&&&&\\ 4&&x&&&&\\ 3&&&x&&&\\ 2&&&&x&&\\ 1&&&&&x&\\ &1&2&3&4&5&6 \end{array}"/></div>
<div> </div>
<div>
<div>205</div>
<div>a) </div>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BA%7D" alt="\overline{A}"/>= Arpakuution heitossa silmä luku on enintään 3</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BB%7D" alt="\overline{B}"/>= Luokasta on ainakin yksi poika</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BC%7D" alt="\overline{C}"/>= Vähintään yhdellä koulun oppilaalla matematiikan arvosna on 10</div>
<span>c) </span>
<div>IV; 0,4</div>
<div><br/>
<div>206</div>
<div>a) P(A)= 2/15</div>
<div>b) P(A)= 4/15</div>
<div>c) P(A)=0</div>
<div>d) P(A)=8/15</div>
<div>e) P(A)=7/15</div>
<div>f) P(A)=15/15</div>
<div> </div>
<div>
<div>208<br/>
5 sinistä, 3 punaista, 4 vihreätä</div>
<div>a) </div>
<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(on%5C%20punainen%5Cright)%3D%5Cfrac%7B3%7D%7B5%2B3%2B4%7D%3D%5Cfrac%7B1%7D%7B4%7D" alt="P\left(on\ punainen\right)=\frac{3}{5+3+4}=\frac{1}{4}"/>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(ei%5C%20punainen%5Cright)%3D%5Cfrac%7B3%7D%7B4%7D" alt="P\left(ei\ punainen\right)=\frac{3}{4}"/></div>
<div>c)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(ei%5C%20ole%5C%20vihre%C3%A4%5C%20eik%C3%A4%5C%20punainen%5Cright)%3DP%5Cleft(%5Cmathrm%7Bon%5C%20%5Csin%20inen%7D%5Cright)%3D%5Cfrac%7B5%7D%7B12%7D" alt="P\left(ei\ ole\ vihreä\ eikä\ punainen\right)=P\left(\mathrm{on\ \sin inen}\right)=\frac{5}{12}"/></div>
<div>d)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(on%5C%20%5Csin%20inen%5C%20TAI%5C%20punainen%5Cright)%3D%5Cfrac%7B5%7D%7B12%7D%2B%5Cfrac%7B3%7D%7B12%7D%3D%5Cfrac%7B8%7D%7B12%7D%3D%5Cfrac%7B2%7D%7B3%7D" alt="P\left(on\ \sin inen\ TAI\ punainen\right)=\frac{5}{12}+\frac{3}{12}=\frac{8}{12}=\frac{2}{3}"/></div>
<div>
<div><br/>
209</div>
<div>4 ja 5, 3 ja 6</div>
<div>4 ja 6, 5 ja 5</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bmatrix%7D%0A6%267%268%269%2610%2611%2612%5C%5C%0A5%266%267%268%269%2610%2611%5C%5C%0A4%265%266%267%268%269%2610%5C%5C%0A3%264%265%266%267%268%269%5C%5C%0A2%263%264%265%266%267%268%5C%5C%0A1%262%263%264%265%266%267%5C%5C%0A%261%262%263%264%265%266%0A%5Cend%7Bmatrix%7D" alt="\begin{matrix} 6&7&8&9&10&11&12\\ 5&6&7&8&9&10&11\\ 4&5&6&7&8&9&10\\ 3&4&5&6&7&8&9\\ 2&3&4&5&6&7&8\\ 1&2&3&4&5&6&7\\ &1&2&3&4&5&6 \end{matrix}"/></div>
<div>Koska todennäköisyys saada silmäluparista summaksi 9 on <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B4%7D%7B36%7D%3D%5Cfrac%7B1%7D%7B9%7D" alt="\frac{4}{36}=\frac{1}{9}"/>, ja summaksi 10 on <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B3%7D%7B36%7D%3D%5Cfrac%7B1%7D%7B12%7D" alt="\frac{3}{36}=\frac{1}{12}"/>.<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B4%7D%7B36%7D%3E%5Cfrac%7B3%7D%7B36%7D" alt="\frac{4}{36}>\frac{3}{36}"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"-->, joten on aina todennäköisempi saada summaksi 9 kuin</div>
<div> </div>
<div>210</div>
<div>a) </div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bmatrix%7D%0A6%26x%26x%26x%26x%26x%26%5C%5C%0A5%26x%26x%26x%26x%26%26%5C%5C%0A4%26x%26x%26x%26%26%26%5C%5C%0A3%26x%26x%26%26%26%26%5C%5C%0A2%26x%26%26%26%26%26%5C%5C%0A1%26%26%26%26%26%26%5C%5C%0A%261%262%263%264%265%266%0A%5Cend%7Bmatrix%7D" alt="\begin{matrix} 6&x&x&x&x&x&\\ 5&x&x&x&x&&\\ 4&x&x&x&&&\\ 3&x&x&&&&\\ 2&x&&&&&\\ 1&&&&&&\\ &1&2&3&4&5&6 \end{matrix}"/></div>
<div>Tämän perusteella:</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(A%5Cright)%3D%5Cfrac%7B15%7D%7B36%7D%3D%5Cfrac%7B5%7D%7B12%7D" alt="P\left(A\right)=\frac{15}{36}=\frac{5}{12}"/></div>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bmatrix%7D%0A6%26%26%26%26%26%26%5C%5C%0A5%26%26%26%26%26%26%5C%5C%0A4%26%26%26%26%26%26%5C%5C%0A3%26%26%26%26x%26x%26x%5C%5C%0A2%26%26%26%26x%26x%26x%5C%5C%0A1%26%26%26%26x%26x%26x%5C%5C%0A%261%262%263%264%265%266%0A%5Cend%7Bmatrix%7D" alt="\begin{matrix} 6&&&&&&\\ 5&&&&&&\\ 4&&&&&&\\ 3&&&&x&x&x\\ 2&&&&x&x&x\\ 1&&&&x&x&x\\ &1&2&3&4&5&6 \end{matrix}"/> </div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B9%7D%7B36%7D%3D%5Cfrac%7B1%7D%7B4%7D" alt="\frac{9}{36}=\frac{1}{4}"/></div>
<div>c)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bmatrix%7D%0A6%26%26%26%26x%26%26%5C%5C%0A5%26%26%26x%26%26%26%5C%5C%0A4%26%26x%26%26%26%26x%5C%5C%0A3%26x%26%26%26%26x%26%5C%5C%0A2%26%26%26%26x%26%26%5C%5C%0A1%26%26%26x%26%26%26%5C%5C%0A%261%262%263%264%265%266%0A%5Cend%7Bmatrix%7D" alt="\begin{matrix} 6&&&&x&&\\ 5&&&x&&&\\ 4&&x&&&&x\\ 3&x&&&&x&\\ 2&&&&x&&\\ 1&&&x&&&\\ &1&2&3&4&5&6 \end{matrix}"/> </div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B8%7D%7B36%7D%3D%5Cfrac%7B2%7D%7B9%7D" alt="\frac{8}{36}=\frac{2}{9}"/></div>
<div><br/>
212</div>
<div>a)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7B365%7D" alt="\frac{1}{365}"/></div>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B12%7D%7B365%7D" alt="\frac{12}{365}"/></div>
<div>c)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B31%2B28%7D%7B365%7D%3D%5Cfrac%7B59%7D%7B365%7D" alt="\frac{31+28}{365}=\frac{59}{365}"/></div>
<div><br/>
214</div>
<div>a)</div>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7B2%7D" alt="\frac{1}{2}"/></div>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7B6%7D" alt="\frac{1}{6}"/></div>
<div>c)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2%7D%7B6%7D%3D%5Cfrac%7B1%7D%7B3%7D" alt="\frac{2}{6}=\frac{1}{3}"/><br/>
<div> </div>
<div>216</div>
<div>
<div>
<div>Vektorit ovat toisiaan vastaan kohtisuorassa täsmälleen silloin, kun pistetulo on 0, eli kun </div>
<img src="https://math-demo.abitti.fi/math.svg?latex=6%5Ccdot%5Cleft(-1%5Cright)%2Bab%3D0" alt="6\cdot\left(-1\right)+ab=0"/></div>
<div>Koska<img src="https://math-demo.abitti.fi/math.svg?latex=6%5Ccdot%5Cleft(-1%5Cright)%3D-6" alt="6\cdot\left(-1\right)=-6"/>, joten ab on oltava 6</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bmatrix%7D%0A6%26x%26%26%26%26%26%5C%5C%0A5%26%26%26%26%26%26%5C%5C%0A4%26%26%26%26%26%26%5C%5C%0A3%26%26x%26%26%26%26%5C%5C%0A2%26%26%26x%26%26%26%5C%5C%0A1%26%26%26%26%26%26x%5C%5C%0A%261%262%263%264%265%266%0A%5Cend%7Bmatrix%7D" alt="\begin{matrix} 6&x&&&&&\\ 5&&&&&&\\ 4&&&&&&\\ 3&&x&&&&\\ 2&&&x&&&\\ 1&&&&&&x\\ &1&2&3&4&5&6 \end{matrix}"/></div>
<div>Siis<img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(vektorit%5C%20ovat%5C%20kohtisuorassa%5Cright)%3D%5Cfrac%7B4%7D%7B36%7D%3D%5Cfrac%7B1%7D%7B9%7D" alt="P\left(vektorit\ ovat\ kohtisuorassa\right)=\frac{4}{36}=\frac{1}{9}"/></div>
</div>
</div>
</div>
</div>
2019-09-27T13:31:39+03:00
Teksti
https://peda.net/id/b2c220b6de8
2019-11-03T23:04:09+02:00
<span>122</span>
<div>A III</div>
<div>B II</div>
<div>C I</div>
<div><br/>
<div>
<div>125</div>
<div>a) 37,5</div>
<div>b) 50</div>
<div>c) 80%</div>
<div>d) 40%</div>
<div>e) 30%</div>
<div> </div>
<div>127</div>
<div>128</div>
<div>129</div>
<div>130</div>
<div>132</div>
<div>135</div>
<div>138</div>
<div>140</div>
</div>
</div>
<div>141</div>
<div> </div>
<div>142</div>
<div>a) [1,6]</div>
<div>b) 3</div>
<div>c) ≈1,51</div>
<div> </div>
<div>145</div>
<div>b)</div>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B719-681%7B%2C%7D29%7D%7B35%7B%2C%7D6%7D%3D1%7B%2C%7D059...%5Capprox1%7B%2C%7D1" alt="\frac{719-681{,}29}{35{,}6}=1{,}059...\approx1{,}1"/>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B59%7B%2C%7D32-57%7B%2C%7D74%7D%7B2%7B%2C%7D45%7D%3D0%7B%2C%7D644...%5Capprox0%7B%2C%7D64" alt="\frac{59{,}32-57{,}74}{2{,}45}=0{,}644...\approx0{,}64"/></div>
<div>Koska 1,1>0,64m, urheilija menestyy paremmin pituushypyssä.</div>
<div> </div>
<div>
<div>146</div>
<div> </div>
<div>148</div>
<div>A</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B15-14%7B%2C%7D5%7D%7B2%7B%2C%7D2%7D%3D0%7B%2C%7D227...%5Capprox0%7B%2C%7D23" alt="\frac{15-14{,}5}{2{,}2}=0{,}227...\approx0{,}23"/></div>
<div>B</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B9%7B%2C%7D4-9%7B%2C%7D7%7D%7B1%7B%2C%7D9%7D%3D-0%7B%2C%7D157...%5Capprox-0%7B%2C%7D16" alt="\frac{9{,}4-9{,}7}{1{,}9}=-0{,}157...\approx-0{,}16"/></div>
<div>0,23>-0,16</div>
<div>A poikkee enemmän kuin B</div>
<div> </div>
<div>150</div>
<div>1,48>1,28</div>
<div>Opiskelija menestyy paremmin ruotsissa.</div>
<div> </div>
<div>152</div>
<div>Välien todellisetarvot</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B155%2B164%7D%7B2%7D%3D159%7B%2C%7D5" alt="\frac{155+164}{2}=159{,}5"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B165%2B174%7D%7B2%7D%3D169%7B%2C%7D5" alt="\frac{165+174}{2}=169{,}5"/></div>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B175%2B184%7D%7B2%7D%3D179%7B%2C%7D5" alt="\frac{175+184}{2}=179{,}5"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B185%2B194%7D%7B2%7D%3D189%7B%2C%7D5" alt="\frac{185+194}{2}=189{,}5"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bx%7D%3D%5Cfrac%7B6%5Ccdot159%7B%2C%7D5%2B15%5Ccdot169%7B%2C%7D5%2B12%5Ccdot179%7B%2C%7D5%2B3%5Ccdot189%7B%2C%7D5%7D%7B%5Cleft(6%2B15%2B12%2B3%5Cright)%7D%3D172%7B%2C%7D8333...%5Capprox172%7B%2C%7D83" alt="\overline{x}=\frac{6\cdot159{,}5+15\cdot169{,}5+12\cdot179{,}5+3\cdot189{,}5}{\left(6+15+12+3\right)}=172{,}8333...\approx172{,}83"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=s%3D%5Csqrt%5B%5D%7B%5Cfrac%7B6%5Ccdot%5Cleft(159%7B%2C%7D5-172%7B%2C%7D83%5Cright)%5E2%2B15%5Ccdot%5Cleft(169%7B%2C%7D5-172%7B%2C%7D83%5Cright)%5E2%2B12%5Ccdot%5Cleft(179%7B%2C%7D5-172%7B%2C%7D83%5Cright)%5E2%2B3%5Ccdot%5Cleft(189%7B%2C%7D5-172%7B%2C%7D83%5Cright)%5E2%7D%7B%5Cleft(6%2B15%2B12%2B3%5Cright)%7D%7D" alt="s=\sqrt[]{\frac{6\cdot\left(159{,}5-172{,}83\right)^2+15\cdot\left(169{,}5-172{,}83\right)^2+12\cdot\left(179{,}5-172{,}83\right)^2+3\cdot\left(189{,}5-172{,}83\right)^2}{\left(6+15+12+3\right)}}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3D8%7B%2C%7D503...%5Capprox8%7B%2C%7D5" alt="=8{,}503...\approx8{,}5"/></div>
<div>
<div><br/>
<span><span>153<br/>
</span></span>
<div>Keksiarvo:</div>
<div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B547%5Ccdot2%2B548%5Ccdot2%2B549%5Ccdot2%7D%7B6%7D%3D548mm" alt="\frac{547\cdot2+548\cdot2+549\cdot2}{6}=548mm"/></div>
<div>Otoskeskihajonta:</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=s%3D%5Csqrt%5B%5D%7B%5Cfrac%7B%5Cleft(548-548%5Cright)%5E2%2B%5Cleft(547-548%5Cright)%5E2%2B%5Cleft(547-548%5Cright)%5E2%2B%5Cleft(548-548%5Cright)%5E2%2B%5Cleft(549-548%5Cright)%5E2%2B%5Cleft(549-548%5Cright)%5E2%7D%7B6%7D%7D" alt="s=\sqrt[]{\frac{\left(548-548\right)^2+\left(547-548\right)^2+\left(547-548\right)^2+\left(548-548\right)^2+\left(549-548\right)^2+\left(549-548\right)^2}{6}}"/></div>
<img src="https://math-demo.abitti.fi/math.svg?latex=%3D0%7B%2C%7D816...%5Capprox0%7B%2C%7D82mm" alt="=0{,}816...\approx0{,}82mm"/></div>
</div>
</div>
2019-09-24T08:53:57+03:00