SOLUTION: 5. You have two biased coins. Coin A comes up heads with probabaility ¼. Coin B comes up heads with probability ¾. However, you are not sure which is which so you choose a coin ran
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Question 578837: 5. You have two biased coins. Coin A comes up heads with probabaility ¼. Coin B comes up heads with probability ¾. However, you are not sure which is which so you choose a coin randomly and you flip it. If the flip is heads, you guess that the flipped coin is B; othervise you guess that the flipped coin is A. Let events A and B designate which coin was picked. What is the probability P(C) that your guess is correct? (Hint: This is an example of sequential experiments, so you might want to use tree diagrams.)
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Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
We know that  = \frac{3}{4})
. We want to find )
. We use Bayes' theorem:
 = P(head|B)\frac{P(B)}{P(head)})
.
We know that P(B) = 1/2 because coins A and B are equally likely to be picked. P(head) is also 1/2 because
 = P(A)P(head|A) + P(B)P(head|B))
 + \frac{1}{2}(\frac{3}{4}) = \frac{1}{2})
Therefore,
 = \frac{3}{4})
Additionally, P(A|tail) = 3/4. The answer is *not* 3/4 + 3/4 = 3/2, that's absurd. It is 3/4 because P(head) and P(tail) both have a 1/2 chance of occurring, and we need to multiply by 1/2 to compensate for each cases.
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