Die A has its six faces marked 1, 2, 2, 3, 3, 3 . Die B is a standard fair die with faces marked 1, 2, 3, 4, 5, 6 . A fair coin is tossed. If it shows Heads, Die A is rolled twice. If it shows Tails, Die B is rolled twice. Given that the sum of the numbers obtained in the two rolls is 5 , the probability that the coin showed Heads is

Step-by-step solution

Let H be the event that the coin shows Heads, and T be the event that it shows Tails. P(H) = 1 2 and P(T) = 1 2 Let E be the event that the sum of the two rolls is 5 . If Heads occurs, Die A is rolled twice. The faces of Die A are 1, 2, 2, 3, 3, 3 . The probability of getting a 2 is P(2) = 2 6 and the probability of getting a 3 is P(3) = 3 6 . The sum is 5 for the outcomes (2, 3) and (3, 2) . P(E|H) = P(2)P(3) + P(3)P(2) = ( 2 6 3 6 ) + ( 3 6 2 6 ) = 6 36 + 6 36 = 12 36 = 1 3 If Tails occurs, Die B is rolled twice.