# Thread: Poker full house problem

1. ## Poker full house problem

In poker a Full House is a hand consisting of 3 cards of one kind and 2 cards of another kind. For example, 'eights and fives' is a hand consisting of 3 eights and 2 fives. hat is the probability of being dealt a full house from a standard deck of 52 cards if the dealing is done fairly?

2. The number of possible fullhouses is 13*12*6*4. See why?.

The total number of 5 card hands is C(52,5).

3. ## not sure?

thanks,i dont exactly see wher the 13 12 6 and 4 come from?could you give me any help?

4. Hello, matty888!

Here ya go . . .

What is the probability of being dealt a Full House?
We want a Triple and a Pair.

There are 13 choices of value for the Triple.
There are: . ${4\choose3} = 4$ ways to get the Triple.

There 12 choices for the value of the Pair.
There are: . ${4\choose2} = 6$ ways to get the Pair.

Hence, there are: . $13 \times 4 \times 12 \times 6 \:=\:3744$ Full Houses.

There are: . ${52\choose5} = 2,598,960$ possible five-card hands.

Therefore: . $P(\text{Full House}) \;=\;\frac{3,\!744}{2,\!598,\!960} \;=\;\frac{6}{4165}$

5. 13 is the # of suits.

6 is the number of combinations possible for the pairs, i.e. 2 cards from 4 choices.

4 is the # of each card type that would get 3 of a kind.

12 is the card types left after one is used up for the 3 of a kind.

So, we have 13*6*4*12=3744