Question

You select a coin at random: 2/3 of the coins are unfair, 1/3 of the coins are fair. The fair coins are equally likely to flip heads or tails. The unfair coins flip heads 3/4 of the times, and tails 1/4 of the times. You flip the selected coin and get heads or tails. Find (1) the probability that the selected coin is fair given the flip is heads, (2) the probability that the selected coin is fair given the flip is tails, (3) the probability that the selected coin is unfair given the flip is heads, and (4) the probability that the selected coin is unfair given the flip is tails.

Answer 1

Let U be the event that the coin is fair and F be the event that the coin is fair

H be the event that the toss resulted in heads and T be the event that the toss resulted in tails

Given P(U) = 2/3, P(F) = 1/3

P(H|F) = P(T|F) = 0.5, P(H|U) = 3/4, P(T|U) = 1/4

a)

Bayes rule

P(H) = P(H|F).P(F) + P(H|U).P(U) Total probability rule

P(H) = 0.5*(1/3) + 0.75*(2/3) = 2/3

Therefore the probability that the coin is fair given the toss resulted in heads is 0.25

b)

Bayes rule

P(T) = P(T|F).P(F) + P(T|U).P(U) Total probability rule

P(T) = 0.5*(1/3) + 0.25*(2/3) = 1/3

Therefore probability that the coin is fair given the toss resulted in tails is 0.5

c) P(U|H) = 1 - P(F|H) = 1 - 0.25 = 0.75

Therefore the probability that the coin is unfair given the toss resulted in heads is 0.75

d) P(U|T) = 1 - P(F|T) = 1 - 0.5 = 0.5

Therefore the probability that the coin is unfair given the toss resulted in tails is 0.5