Suppose you have a standard 52-card deck of playing cards. Let\'s develop the li

Question

Suppose you have a standard 52-card deck of playing cards. Let's develop the likelihood that we are dealt a three of a kind, (adapted from [Sullivan 2007]) How many ways can five cards be selected from a 52-card deck? Each deck contains 4 twos, 4 threes, and so on. How many ways can three of the same card be selected from the deck? The remaining two cards must be different from the three chosen and different from each other. For example, if we drew three kings, the fourth card cannot be a king. Further, if we drew three kings, the fourth and fifth cards cannot be the same (or we'd have a full house). After selecting the three of a kind, there are 12 different ranks of card remaining in the deck that can be chosen. Of the 12 ranks remaining, we choose two of them and then select one of the four cards in each of the two chosen ranks. How many ways can we select the remaining two cards? Compute the probability of obtaining three of a kind when dealt five cards.

Explanation / Answer

a) 5 cards can be drawn from a a well-shufled pack of 52 cars in 52C5 ways = 2598960

b) The number of ways in 3 of the same card be selected from the deck is = 4C3 = 4. There are 13 different ranks of cards. So total number of ways = 13*4 = 52

c) The remaining 2 cards are choosen different ranks and also other rank of the rank of selected 3 cards.

So we have 12 different ranks of cards. Select 2 ranks out of 12 ranks is 12C2 = 66

Select 1 card from one type of rank cards(4) and another card from another typ of rank cards(4) = 4*4 = 16

Total number of ways = 16*66= 1056

d) Required probabilty = (1056*52) / 2598960

=0.02113

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