A pocket contains three coins one of which had a head on bot

A pocket contains three coins, one of which had a head on both sides, while the other two coins are normal. A coin is chosen at random from the pocket and tossed three times. What is the probability of obtaining three heads?

Solution

There are two cases here:

1. Getting a normal coin, and getting 3 heads.
2. Getting the abnormal coin, getting 3 heads.

For Case 1:

The probability of 3 heads given that it is a normal coin is

P(3 heads|normal) = (1/2)(1/2)(1/2) = 1/8 = 0.625

For case 2:

The probability of 3 heads given that it is an abnormal coin is

P(3 heads|abnormal) = 1*1*1 = 1

Thus, from Bayes\' Rule,

P(3 heads) = P(normal)P(3 heads|normal) + P(abnormal)P(3 heads|abnormal)

= (2/3)(1/8) + (1/3)(1)

= 5/12 or 0.416666667 [ANSWER]