If five cards are schosen at random from a conventional 52 card deck, what is the probcapacity of getting a complete residence.

You are watching: The probability of being dealt a full house

This is what I am reasoning.$(52*inom43*inom42)/_52C_5$

Is that right?


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A complete house has actually three cards of one kind and also two of an additional, so think around it favor this: initially you pick a form of card (13 choices), then you pick three out of 4 of those cards, then you select a second type of card, and finally you select 2 of those four cards. Therefore you have $13choose 14pick 312choose 14choose 2$ feasible complete house hands. So the probcapacity is then

$$13pick 14select 312choose 14choose 2over52select 5=(13)(4)(12)(6)over2598960=3744over2598960approx0.00144$$


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Arthur Skirvin"s answer is great, but I simply desire to administer one more reasoning process for the numerator:

$$( extrmselect 2 kinds)cdot( extrmchoose the type of three cardscdot extrmchoose cards)cdot( extrmselect the type of cardscdot extrmselect cards)\= 13choose2cdot2choose14choose3cdot1choose14choose2.$$


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You might also think around it this means, wbelow I assume the card choices to be order dependent in both the numerator and the denominator.

The full variety of possible choices is $52 imes51 imes50 imes49 imes48$.

To find the variety of complete residence selections, initially pick three out of the 5 cards. For the 3 cards you have actually $52 imes3 imes2$ cases (when you pick the first card, the remainder need to have actually the exact same number), and also for the two cards you have actually $48 imes3$ cases (you cannot pick from the previously selected class).

Thus the probability would be$fracC(5,3) imes52 imes3 imes2 imes48 imes352 imes51 imes50 imes49 imes48approx0.00144$.


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Not fairly. I would certainly break the counting up into the adhering to procedures.

How many means are there to pick the face worth for the triple?How many means are tbelow to select the suits for the triple? (You currently have this.)How many type of means are there to pick the challenge value for the pair?How many kind of means are tright here to pick the suits for the pair? (You already have actually this.)
Unfortunately, it"s a small difficult to parse your question without LaTeX formatting, which compounds through the fact that I never use that specific notation for binomial coefficients. For those that don"t know, a full home is a hand of $5$ cards such that $3$ of them share the very same rank and the remaining $2$ likewise share the same rank.

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You have actually $13$ choices (2-10, J, K, Q, A) for the rank of the triple, and also once that has been chosen, you have actually $12$ remaining choices for the rank of the pair. Remember you additionally have to choose $3$ suits of the feasible $4$ for the triple, and also $2$ of the feasible $4$ for the pair. Expush this indevelopment in terms of binomial coefficients to acquire the complete variety of possible full residence hands.

Then sindicate divide by the full variety of unlimited $5$-card hands for the probcapability.