Problem 7 (Combinations Poker Probabilities) Suppose you deal a poker hand of 5

Question

Problem 7 (Combinations Poker Probabilities) Suppose you deal a poker hand of 5 cards from a standard deck as discussed in lecture. (a) What is the probability of a flush (all the same suit) of all red cards? (b) What is the probability of a full house where the 3-of-a-kind include two black cards, and the 2- of-a-kind are not clubs? (c) What is the probability of a single pair, where among the 3 non-paired cards, we have 3 distinct suits? (d) What is the probability of a pair, where 3 of the cards are black and 2 are red? Hint: These are straight-forward modifications of the formulae given in lecture, except for (d). For that, think about the two different parts of the problem, and observe that they are independent (the colors are independent of the ranks/denominations). You may quote formulae already given in lecture or lab without attribution and without explaining how to get the formula.

Explanation / Answer

Part a)

For flush , there can be any two red suits

The number of ways of selecting 5 cards from one suit = 13C5

Total number of ways of selecting 5 cards from a deck = 52C5

Hence total probability of red flush = 2*13C5 / 52C5

P(red flush) = (2 * 13*12*11*10*9 / (5*4*3*2*1)) / (52*51*50*49*48/(5*4*3*2*1))

P(red flush) = 0.00099

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