There are 3 coins in a box. One is a two-headed coin; another is a fair coin; and the third is a biased coin that comes up heads 75 percent of the time. When one of the 3 coins is selected at random and flipped, it shows heads. What is the probability that it was the two-headed coin?

Answer: 4/9

Explanation:

C1 : coin with 2 heads is chosen

C2: the fair coin is chosen

C3: the 3rd coin is chosen

H1: coin flips result in the head.

The probability of when a coin is randomly chosen:

P (C1) / (C2) / (C3) = 1/3

Probability of each coins showing head is:

P (H/C1) = 1

P (H/C2) = 1/2 = 0.5

P (H/C3) = 3/4 = 0.75

Bayes theorem's describes the probability of an event based on prior knowledge of conditions that have a relationship with the event.

The probability that it was a two-headed coin is P (C1/H)

Using Bayes formula:

P (C1/H) = P (H/C1) P (C1) / P (H/C1) P (C1) + P (H/C2) P (C2) + P (H/C3) P (C3)

Which is

(1*1/3) / (1*1/3) + (0.5*1/3) + (0.75*1/3)

=4/9

Therefore, the probability that it was a two-headed coin is 4/9.