A poker deck consists of cards ranked 2,3,4,5,6,7,8,9,10,J, Q,K, A, (13 different ranks) each in four suits, for a total of 52 distinct cards. A pinochle deck consists of two copies each of cards ranked 9,10,J, Q,K, A, each in four suits, for a total of 48 cards (24 pairs of identical cards).

(a) What is the probability that a five-card poker hand drawn from a pinochle deck contains no duplicate cards?

(b) Suppose I select a deck of cards at random, and that deck has a 50% chance of being a pinochle deck and a 50% chance of being a poker deck. I then deal a random five-card poker hand and give it to you. You look at the hand and determine that it could have come either from a poker deck or a pinochle deck. In other words, the hand you are dealt contains no duplicates and it only has cards ranked 9 or higher. Use Bayes’ Theorem to determine the probability that I drew from a pinochle deck.