# Thread: Probability of Poker Hands

1. ## Probability of Poker Hands

I am stuck on the following question:
A poker hand consists of 5 cards dealt from an ordinary deck of 52 playing cards.
a) How many poker hands are possible?
b) How many different hands consisting of 3 kings and 2 queens are possible? c) A full house consists of 3 cards of one denomination and 2 cards of another. How many different full houses are possible?
d) Calculate the probability of being dealt a full house.

On "a" I am stuck on 2 different answers. My first answer I figured c1*c1-1*c1-2*c1-3*c1-4= 52*51*50*49*48= 311,875,200 because I figured every time you dealt a card, there was one less card to be dealt in the deck. However, looking around different sites I keep seeing that the answer to part "a" is c(52,5) = 2,598,960. So now I don't know which is correct.
As for parts b-d I'm completely lost!

Thanks again for any help!!!

2. Originally Posted by Shanynn

I am stuck on the following question:
A poker hand consists of 5 cards dealt from an ordinary deck of 52 playing cards.
a) How many poker hands are possible?
b) How many different hands consisting of 3 kings and 2 queens are possible? c) A full house consists of 3 cards of one denomination and 2 cards of another. How many different full houses are possible?
d) Calculate the probability of being dealt a full house.

On "a" I am stuck on 2 different answers. My first answer I figured c1*c1-1*c1-2*c1-3*c1-4= 52*51*50*49*48= 311,875,200 because I figured every time you dealt a card, there was one less card to be dealt in the deck. However, looking around different sites I keep seeing that the answer to part "a" is c(52,5) = 2,598,960. So now I don't know which is correct.
As for parts b-d I'm completely lost!

Thanks again for any help!!!
a) you are counting each permutation of a hand as a distinct hand when
they are in fact the same hand.

Therefore the number of distinct hands is 311,875,200/(5!)= 2,598,960.

RonL

3. I feel like such an idiot, but I still can't figure out how to get 2,598,960 as my answer. What are the steps that lead to that answer? Probabilities is driving me crazy!!!!

4. Originally Posted by Shanynn
I feel like such an idiot, but I still can't figure out how to get 2,598,960 as my answer. What are the steps that lead to that answer? Probabilities is driving me crazy!!!!
there are 52 cards in the deck. you want to find the number of ways you can choose 5 cards from those. hence it is the combination $_{52}C_5 = {52 \choose 5} = 2598960$

5. ## Finally get it!

Thanks... I think I finally understand. So basically it's 52!/5! 47! so I multiply 52x51x50x49x48x47!, then cancel out both 47!'s, which gives me 311,875,200 and then I divide by 5! (5x4x3x2x1) (120) and voila! It's 2,598,960. Thanks again for the help! I forgot about the cancellation so I was making it extra hard by doing all of these problems the long way (52x52x50x49x48x47x46x46x44........x2x1) Talk about frustrating!