Hey Guys,
Last week one night I was playing a 1-2 NL holdem cash game at the bar. I heard someone blurt out "330!" Then another man said "250". I look up at the T.V. to see what they were yelling about. I seen a trivia question from the Full Tilt Poker show. The question was, "What are the odds that you are dealt pocket aces?" I laughed and then blurted out 220 to 1. Of course I was right, but I was amazed that these guys didn't know the answer. These guys are fairly good players too. They asked, "how did I know that"? I explained it to them in a formula sense. Then it dawned on me that how many people don't understand their Combinations for poker math.
Example.
The other day I played 5 card draw. Each player is dealt a 5 card hand from a 52 card deck. I wanted to figure out how many possible 5 card hands there are.
No problem. There are 52 possibilities for the first card, then 51 possibilities left for the second card. 50 possibilities for the third card, then 49 possibilites, then 48 possiblilities. So the total number of hands is
52 x 51 x 50 x 49 x 48 = 311,875,200
There is something wrong. What we have here is a permutation. We will have some hands that have the same exact cards.
Example of a permutation --- See below we have the exact same cards just in a different order.
7 of hearts, king of hearts, 10 of spades, 3 of diamonds, ace of clubs
king of hearts, 7 of hearts, 10 of spades, 3 of diamonds, ace of clubs
We are wanting different combinations of hands. Number of different ways of chosing the 5 cards in our hand.
We realize that there are 120 ways of putting 5 objects in order.
5 x 4 x 3 x 2 x 1 = 120 another permutation.
The number 311,875,200 is 120 times to big to be considered a combination. Therefore, the number of 5 card hands, when you don't want to count each different ordering seperately, must be:
52 x 51 x 50 x 49 x 48 -------311,875,200
__________________ = _______________ = 2,598,960
5 x 4 x 3 x 2 x 1-------- ----------120
The numbers of ways of selecting objects in this way is called the number of Combinations of X objects taken Y at a time. When you count the number combinations, the order of the objects what you select does not matter. Notice that the number of combinations is therefore less than the number of permutations.
Combinations are expressed as:
(X)
(Y)
(X)
(Y) : This expression is sometimes read as "X choose Y," because it tells you the number of different ways of choosing Y objects from a group of X objects.
One more small Example.
I am a coach of a baseball team. I only have 9 players on my team. Just enough to fill my batting order. I don't know which order I want my players to bat. I am going to figure out how many different ways I can order my roster.
9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362,880 This is another permuatation, not combinations
Okay, Now I am going to figure out how many different combinations are there in my roster.
Our formula is expressed as :
(X) Objects Taken from
(Y) At a time
(9) Objects Taken from
(9) At a time
9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1----362,880
________________________ = ___________ = 1
9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1----362,880
Now, If I had 12 players to fill my 9 ordered roster it would look like this
(12) Objects Taken from
(9) At a time
12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4------79,833,600
___________________________ = ________________ = 220
9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1------------362,880
Do you see how these 3 different people on the same 9 order batting roster made a big difference?
Okay, I hope you learned something here. Next posts will be how to use these combinations into starting hand formulas. Feel free to ask questions - email me or post a comment.
