There were three other solutions submitted. Correct solutions were also received from Kevin Bourrillion and Jeff Decker.
Answer: 1/3, 1/5, 1
a) There are two events: The selection of a coin -- resulting in the Unfair Coin, "U", or the Fair Coin, "F" -- and the flip -- resulting in heads, "H", or tails, "T", for the Fair Coin, and in one of the two heads, "H1" or "H2", for the Unfair Coin. Each of these events is considered a 50/50 proposition.
Event1 Event2 F H F T U H1 U H2So the conditional probability P(F|H) = P(F and H)/P(H) = (1/4)/(3/4) = 1/3.
b) Similarly,
Event1 Event2 Event3 F H H F H T F T H F T T U H1 H1 U H1 H2 U H2 H1 U H2 H2Now P(F|H,H) = P(F and H,H)/P(H,H) = (1/8)/(5/8) = 1/5.
c) It must be the fair coin. P(F|H,H,T) = 1.
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