Question

Suppose you are given a fair coin and an unfair.com, which

always comes out Head with probability 2/3. You chose one
at randoms of the Coins, tossed it twice and the outcomes were
both Heads. What is the probability that the com is fair​

Answer 1

9/25 or 36%

Explanation:

The probability of selecting the fair coin is 1/2, and the probability of getting heads with the fair coin is 1/2.

P(fair) = 1/2, P(heads | fair) = 1/2

The probability of selecting the unfair coin is 1/2, and the probability of getting heads with the unfair coin is 2/3.

P(unfair) = 1/2, P(heads | unfair) = 2/3

P(fair and 2 heads) = 1/2 (1/2)²

P(fair and 2 heads) = 1/8

P(2 heads) = 1/2 (1/2)² + 1/2 (2/3)²

P(2 heads) = 1/8 + 2/9

P(2 heads) = 25/72

Therefore, the probability that you selected the fair coin, given that you flipped 2 heads, is:

P(fair | 2 heads) = (1/8) / (25/72)

P(fair | 2 heads) = (1/8) (72/25)

P(fair | 2 heads) = 9/25