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We have a standard deck of 52 cards, with 4 cards in each of 13 ranks. We call a 5-card poker hand a full house if the hand has 3 cards of one rank and 2 cards of another rank (such as 33355 or AAAKK). What is the probability that five cards chosen at random form a full house?

Sep 9, 2021

#1
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You choose a rank for the three cards, then choose a rank for the two cards.  There are C(52,5) ways of choosing 5 cards, so the probability is

$\frac{\binom{13}{2} \binom{4}{2} \binom{4}{3}}{\binom{52}{5}} = \frac{3}{4165}.$

Sep 9, 2021
#2
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I'm actually not to familiar with statistics but would be very happy to learn where you learned this, do you think you'd be able to point me to a resource?

helperid1839321  Sep 9, 2021
#3
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Guest’s answer is wrong. So it appears he didn’t learn it.

There are numerous examples of online poker-hand probability texts and examples. Google is a good starting point ...Bing is too.... so is Yahoo.... and Dogpile....

Solution for this question:

Choose the rank for the triple nCr(13,1)

Choose three suits for the triple nCr(4,3)

Choose a different rank for the pair nCr(12,1)

Choose two suits for the pair nCr(4,2)

Then

Divide this by the total number of five-card hands. nCr(52,5)

$$\hspace {7cm} \large \text{ } \dfrac{\dbinom {13}{1}\dbinom {4}{3}\dbinom {12}{1}\dbinom {4}{2}}{\dbinom {52}{5}} = \dfrac{6}{4165} \approx 0.144\%$$

GA

Sep 10, 2021
edited by Guest  Sep 10, 2021