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(a)A gambler has a fair coin and a two-headed coin in his pocket. He selects one of the coins at random; when he flips it, it shows heads. What is the probability that it is the fair coin? (b) Suppose that he flips the same coin a second time and, again, it shows heads. Now what is the probability that it is the fair coin?

answer to (a) is 1/3 which you need for (b), the answer to (b) is

answer

I learned the basics of Bayes, but I don't understand what it means to have $O_1$ and $O_2$

Problem (c)) Suppose that he fluids the same coin a third time and it shows tails. What's the probability that it is the fair coin? How do we solve this?

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    For part (c), the answer very obviously should be $1$ since $HHT$ cannot be observed with the two-headed coin. But you can proceed systematically using $P(HHT|F) = 1/8$, $P(HHT|F^c) = 0$ and plug and chug in $$P(F|HHT)=\frac{P(HHT|F)P(F)}{P(HHT|F)P(F)+P(HHT|F^c)P(F^c)}.$$2012-09-18
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    @DilipSarwate what page can i find a usage instruction for the notations you're using?2012-09-19

3 Answers 3