# Math Help - Probability in cards

1. ## Probability in cards

I am having trouble with a card game called. Five card brag
(three card brag is the game that is played in Lock Stock and Two Smoking Barrels)

In this game you are dealt five cards. After looking at your cards you ditch two of them and play with the remaining three.

My question is simple. What are the odds of drawing the following hands with five cards?

3 of a kind
three card straight
three card flush
three card straight flush
one pair
high card

assume a 52 card deck with no wilds.

2. If you are dealt 5 cards, the probability of getting 3 of a kind:

You choose 3 cards from 4 of the same. i.e. 3 aces from the 4 aces, 3 kinds from the 4 kings, etc. There are 13 suits.
So, we have 13C(4,3)
But, we have to choose 2 cards from the remianing 48, so we have:

$13C(4,3)C(48,2)=58656$

There are C(52,5)=2,598,960 different possible combos, so the probability of getting a 3-of-a-kind is $\frac{58656}{2598960}=\frac{94}{4165}\approx{.022}$

But...one little thing to mention. This also includes the fullhouse. Afterall, a fullhouse is a fancy three of a kind. To cut to the quick, there are 3744 different fullhouses. You can subtract this if you wish.

Try to count up some of others. Get out a deck and think it through. It's rather fun.

3. ## i see

well it all sounds good and from what I can tell it is correct. it does match some data I found on a few card playing sites

Unforunatly, for me, I undertand very little about statistical math. in fact i know nothing. so when you write something like
C(52,5)=2,598,690

telling me how many five card hands are possible. I dont really understand it.

can you explain it to me? talk to me as you would a small child. If explained I WILL understand I assure you. I just need a little more to work with

4. C(52,5) means how many ways can we choose 5 items from 52. Intuitively, we know it's going to be rather large. That is called a combination. If order matters it is called a permutation. A combination lock would be a permutation because order matters. That is why they should be called permutation locks and not combination locks.

The formula for $C(n,r)=\frac{n!}{(n-r)!r!}$

Those exclamation points do not mean we are excited about the r's and n's. That means factorial. A factorial is a number multiplied by each consecutive integer down to 1. For instance, 10! would be 10*9*8*7*6*5*4*3*2*1
You can see where 52! would be huge.

It is more involved than I can explain it all here. Google something on it and there will be plenty.

5. For the 3 card straight flush. With a five card hand, there are 40 possible straight flushes. That includes the royal flush. So, with 3 cards, there are 48 possible straight flushes.

A,K,Q K,Q,J Q,J,10, J, 10, 9 10 9 8 9,8,7

8,7,6 7,6,5 6,5,4 5,4,3 4,3,2 2,A,K

There are 4 suits, so 12*4=48.