The game of Pinochle is played with a deck of 48 cards. However, the cards are n

Question

The game of Pinochle is played with a deck of 48 cards. However, the cards are not all unique; in fact there are two copies of every card. That is, we can consider the deck of consisting of cards numbered from 1 to 24 and with two cards numbered i for every i = 1, 2, . . . 24. There are 4 players in Pinochle and each player is dealt 12 cards.

a) My father once called me and told me that he had been dealt a hand that consisted entirely of doubles. He asked me what the probability of this happening was. Calculate the probability that player 1 is dealt a hand of all doubles.

b) Often when people see a surprising event they actually ask the wrong question. It would have been surprising if any of the 4 players had been dealt a hand entirely of doubles. Compute the probability that at least one of the players was dealt a hand entirely of doubles.

Explanation / Answer

We need to select only 6 cards and the other cards are automatically selected.

Number of ways of selected 6 cards = 24*23*22*21*20*19

SIne ordering is not important . No.of ways = 24*23*22*21*20*19 / 2^6

Probability =  24*23*22*21*20*19 / 2^6 / [ 48C12] =  24*23*22*21*20*19 / 2^6 * 36! * 12! / 48! = 0.00002173

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