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?

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.