Question 1194840
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Define these events:<ul><li>A = a white ball was selected from the first urn. It is transferred to the second urn.</li><li>B = a black ball was selected from the first urn. It is transferred to the second urn.</li><li>C = a black ball was selected from the second urn</li></ul>Then we can form these probabilities:<ul><li>P(A) = 5/12 since there are 5 white out of 5+7 = 12 total in the first urn</li><li>P(B) = 7/12 since there are 7 black out of 5+7 = 12 total in the first urn</li><li>P(C given A) = 9/13 because there are 9 black balls and 3white+9black+1extraWhite = 13 balls total in the second urn. Don't forget to add on that extra 1 white ball</li><li>P(C given B) = 10/13 for similar reasoning as the previous probability. This time we have 1 extra black ball to get 9+1 = 10 total in the second urn.</li></ul>The keyword "given" refers to conditional probability. It indicates that the prior event in question is known to already have happened.


Now use the law of total probability
P(C) = P(C and A) + P(C and B)
P(C) = P(C given A)*P(A) + P(C given B)*P(B)
P(C) = (9/13)*(5/12) + (10/13)*(7/12)
P(C) = (9*5)/(13*12) + (10*7)/(13*12)
P(C) = 45/156 + 70/156
P(C) = (45+70)/156
P(C) = 115/156
I would leave it in fraction form since it's the most exact.


If your teacher requires decimal form, then,
115/156 = 0.737179 approximately
Round this however your teacher instructs
This converts to 73.7179% approximately



Answer in fraction form is <font color=red>115/156</font> (exact)
Answer in decimal form is <font color=red>0.737179</font> (approximate)
Answer in percent form is <font color=red>73.7179%</font> (approximate)


Further reading about the law of total probability
<a href = "https://www.probabilitycourse.com/chapter1/1_4_2_total_probability.php">https://www.probabilitycourse.com/chapter1/1_4_2_total_probability.php</a>


Further reading about conditional probability
<a href = "https://www.mathsisfun.com/data/probability-events-conditional.html">https://www.mathsisfun.com/data/probability-events-conditional.html</a>
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