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Name:  brian
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1995


Question:
I am doing a science project on probability of poker hands. Do you have any background information for this?


Replies:
For a reference I would suggest a text on probability that discusses the topics "combinations" and "permutations". You will need to understand those terms, as well as the concept of probability, in order to follow the rest of this. I will show you how to do one of the calculations, then you try to do the others. Technically, we will be computing the probability of being dealt a hand of 5 cards having exactly 1 pair (no discard & draw).

First, some notation. N! (read "N factorial") means the product

N * (N - 1) * (N - 2) * . . . * 3 * 2 * 1, so for example

5! = 5 * 4 * 3 * 2 * 1 = 120. Also, the symbol /M\ M!

| | = --------- and is often read "M choose N". The probability of getting \N/ N! * (M - N)!

a one-pair hand = (# of hands containing one pair)/(total # of hands).

The total number of hands is:
 
 /52\   52*51*50*49*48
 |  | = -------------- = 2,598,960
 \5 /      5*4*3*2*1
 

Let us construct, in words, a one-pair hand: of the 13 values (2 through Ace), choose 1; of the 4 cards of this value, choose 2 (the pair); of the remaining 12 values choose 3 (we need 3 different values so we do not get another pair or 3 of a kind); of the 4 cards of each of these 3 values, choose 1.

In symbols,
 /13\ /4\ /12\ /4\ /4\ /4\
 |  | | | |  | | | | | | | = 13*6*220*4*4*4 = 1,098,240
 \1 / \2/ \3 / \1/ \1/ \1/
 

the number of hands containing exactly 1 pair. Thus, the probability of being dealt a one-pair hand = 1098240/2598960, or about 0.42257.

A couple of notes: the truly awful-looking
   /M\ 
   | |
   \N/

is the best I could do toapproximate the standard notation:

/     \
 |     | 
 \     /  
are supposed to be big left and right parentheses, respectively.

When I computed the 52!/(5! * 47!) I used a common "trick" to reduce calculation effort: note that 52! = 52 * 51 * 50 * 49 * 48 * 47!, and that last term 47! cancels the 47! in the denominator (5! * 47!). However, some calculators have a factorial button, making this trick unneeded.

When you calculate the number of ways of getting a straight, for example, the easiest way is to include the straight flushes. Then calculate the number of straight flushes and subtract (if desired) to get the number of non-flush straights. Through all this, though, keep in mind that the most important goal of this project is for you to gain some understanding of probability! Without that, doing the calculations is kind of pointless.

rcwinther


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