Ask A Scientist Mathematics Archive

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

Response #:  1 of 1
Author:      rcwinther
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
                    /52\   52*51*50*49*48
total # of hands is |  | = -------------- = 2,598,960 for the denominator.
                    \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.
            /13\ /4\ /12\ /4\ /4\ /4\
In symbols, |  | | | |  | | | | | | | = 13*6*220*4*4*4 = 1,098,240 = the
            \1 / \2/ \3 / \1/ \1/ \1/
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\ is the best I could do to
                                             | |
                                   /     \   \N/
approximate the standard notation: | and | are supposed to be big left
                                   \     /
and right parentheses, respectively. You may also see the notation    C
(the M and N are subscripts) for "M choose N".                      M   N

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.