In a poker hand consisting of 5 cards, find the probability of holding 4 hearts and 1 club.

Question

Volosoyx · Accepted Answer

There are total 13 hearts and 13 clubs in the standard 52 card deck.
The number of ways to choose 4 out of 13 hearts is
$(\begin{matrix} 13 \\ 4 \end{matrix}) = \frac{13!}{4! (13 - 4)!} = 715 w a y s$
The number of ways to choose 1 out of 13 clubs is
$(\begin{matrix} 13 \\ 1 \end{matrix}) = \frac{13!}{1! (13 - 1)!} = 13 w a y s$
If the first operation can be performed in 715 ways (the number of ways to choose 4 out of 13 hearts), and for each of these ways the second operation can be performed in 13 ways (the number of ways of choosing 1 ot of 13 clubs), then these 2 operations can be performed together in
715*13=9295 ways
The number of ways to choose 5 out of 52 cards (a poker hand) is 2598960.
Hence, the probability of choosing 4 hearts and 1 club in a poker hand is
$P = \frac{T h e n u m b e r o f w a y s t o c h o o s e 4 h e a r t s a n d 1 c l u b}{T h e n u m b e r o f w a y s t o c h o o s e a p o k e r h a n d} = \frac{9295}{2598960} = 0.0036$
Result:
0.0036

In a poker hand consisting of 5 cards, find the

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