5. A hand in poker consists of 5 cards. Suppose 5 cards are dealt from a well-sh

Question

5. A hand in poker consists of 5 cards. Suppose 5 cards are dealt from a well-shuffled deck (52 cards. 4 suits.O. O values are 2-10, J, Q, K. A), without replacement. (a) What is the chance of there being exactly 2 hearts in the hand? (b) A flush in poker is 5 cards of the same suit (all spades, all hearts, all diamonds, or all clubs), but the five cards cannot be consecutive (e.g., 2, 3, 4, 5, 6 of hearts is a straight flush, not a flush). What is the chance of being deal a flush from a well-shuffed deck of cards? (Hint: It might be easier to include the consecutive ones and then subtract them from the total.)

Explanation / Answer

5 cards from a deck of 52 cards can be selected in 52C5 ways

a)
2 hearts card can be selected in 13C2 ways
Remaining 3 cards from 39 cards can be selected in 39C3

Required probability = 13C2*39C3/52C5 = 0.2743

b)
5 cards of the same suit, possible combinations are 4C1*13C5
There are 12 possible outcome of a straight flush in any suit i.e. 12*4 = 48 straight flush are possible

Hence, possible combinations of flush but not straight flush are 4C1*13C5 - 48 = 5100

Required probability = 5100/52C5 = 0.0020

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