# What is the probability of

1. May 18, 2009

What is the probability of....

1. Ten cards are dealt from a deck of 156 cards (3 standard 52 card decks) What is the probability of getting at LEAST 3-of-a-kind (up to 10-of-a-kind).

My Attempt:
3. P(at least 3-of-a-kind)
= 1 - P(10 different ranks) - P(9 different ranks)
= 1 - [13C10 * (12C1)^10 + 13C9 * (9C1) * (12C2)^1 * (12C1)^8] / [156C10]
= 1 - [ 286 * 12^10 + 715 * 9 * 66 * 12^8 ] / 1752195368913990
= 1 - 200325892276224 / 1752195368913990
= 1 - 0.1143285137207
= 0.8856714862793

2. May 18, 2009

### larstuff

Re: What is the probability of....

I'm not quite sure what the question is here. What does 3-of-a-kind mean in this situation? Does it mean 3 of the ace of Spades, 3 spades, 3 aces...
In other words, precicely what is the outcome we seek to get?

And - anyway, do not forget all possible combinations in your calculation.

3. May 18, 2009

Re: What is the probability of....

3-of-a-kind is like 3 Fours, 3 Fives, etc... I need help either finding the probability to getting 3-of-a-kind if 10 cards are dealt or AT LEAST 3-of-a-kind. I hope that clears it up.

4. May 18, 2009

### larstuff

Re: What is the probability of....

Yes, it does. There are 3x4 of each kind then in the three decks, that's to say 12 Fours, 12 Fives and so on. This is not a simple task, as far as I can see. Try to rule out the "oposite", that's to say, find the probability of getting only 1 of each kind + 1 or 2 of each kind + 2 of each kind. If this is possible, the rest will be the probability of getting 3 or more of a kind.

Last edited: May 18, 2009
5. May 18, 2009

Re: What is the probability of....

Exactly, there are 12 of each kind and 10 cards are being dealt.

6. May 19, 2009

### CompuChip

Re: What is the probability of....

I assume you want the total probability, so it suffices to calculate
1 - P
where P is the probability "there are at most 2 of every card".

Then because all cards are equivalent, I'd first calculate the probabilities of
2,3,4,5,6,7,8,9,10,J
2,3,4,5,6,7,8,9,10,10
2,3,4,5,6,7,8,8,9,9
2,3,4,5,6,6,7,7,8,8
2,3,4,4,5,5,6,6,7,7
2,2,3,3,4,4,5,5,6,6

Then you can rename the cards (for example, in the first one I have chosen 2 - J but instead of those 10 you can have any other, so you'd get a factor binom(13, 10) I think, similarly for the others).

Could this work or did I miss any important points?