A box contains 5 coins: 4 regular coins and 1 fake coin. When a regular coin is tossed, the probability P(head) = 0.5, a...

Question

A box contains 5 coins: 4 regular coins and 1 fake coin. When a regular coin is tossed, the probability P(head) = 0.5, and for a fake coin, P(head) = 1. You pick a coin at random and toss it twice, and get two heads. The probability that the coin you have chosen is the fake coin is __________. (rounded off to two decimal places)

Accepted Answer

P(fake coin | HH) = 0.50 You randomly pick one of 5 coins (4 fair, 1 always-heads). After tossing twice and getting HH, you update belief using Bayes theorem. The fake coin always gives HH (probability 1), while a fair coin gives HH with probability 0.25. The posterior probability of having the fake coin is (1/5 * 1) / (1/5 * 1 + 4/5 * 1/4) = 0.2 / 0.4 = 0.5.