Question

5. In poker, a “flush” is a five-card hand where all five cards have the same suit. A hand has “three of a kind” when any three cards have the same rank. “Four of a kind” is defined analogously. A flush beats a three of a kind, but is beaten by four of a kind. Demonstrate that this ranking of hands corresponds to how unlikely each hand type is to occur when drawing five cards at random (Hint: just count the number of different hands of each type, and then rely on the principle of exchangeability).

Answer 1

Number of ways of selecting 5 cards out of 52 cards is

Four of a kind:

There are total 13 denominations and each denomination has 4 cards. So number of ways of selecting 1 denominations and then 4 cards out of 4 is

And since we need 4 of same kind so remaining 1 card must come from different denomination so number of ways of selecting 1 denominations out of remaining 12 denominations and then 1 card from selected denomination is

So number of ways of selecting four of a kindis :

So probability of getting four of a kind:

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3 of a kind:

Number of ways of selecting 1 denominations out of 13 is C(13,1). Number of ways of selecting 3 cards out of 4 cards of selected denomination is C(4,3). And then select two denominations out of remaining 12 denominations is C(12,2) and then 1 card from each selected denominations is C(4,1)C(4,1). So number of ways are there to draw a 5 card poker hand that contains 3 a kind is

C(13,1)C(4,3)C(12,2)C(4,1)C(4,1) = 54912 ways

So probability of getting three of a kind:

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Flush:

There are total 4 suits and each suit has 13 cards. So number of ways of selecting 1 suit and then 5 cards out of 13 is

So probability of getting flush:

That is out of these it is most likely to get three of a kind, then flush and then four of a kind.