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Probability and Poker
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Probability and the Flush

The flush is just a bit easier to get than a full house. It is a hand well worth trying for in a poker game. To find out what the chances are of being dealt a flush we must resort to a bit of mathematics.

(4 X 1287) - 40 = 5108

The flush in poker is comprised of five cards all in the same suit.The 4 in our equation comes from the number of suits. The 1287 is the number of combinations possible of the thirteen cards within each suit. Now, in previous hands the number of possibilities have been small enough that we could simply lay them out on a piece of paper. But 1287 combinations is more than most people want to puzzle through. But there is a simple mathematical formula for getting this number. It goes something like this:

n!/((n-r)! X r!)

"Okay," you may ask, "what are these excamation points?" The answer is very simple ... they are factorials. And factorials are the multiplication of a bunch of numbers in a series. For example, 4! = 4 X 3 X 2 X 1. The "n" in our equation is the number of different cards to choose from within a set. And the r is the number of items we are trying to find. So this is how we got 1287:

13 X 12 X 11 X 10 X 9 / 5 X 4 X 3 X 2 X 1 = 1287

The (n - r)! part of the equation allows us to get rid of the lower end of our n! numerator. What we are really doing is mathematically mimicking reality in our choices. Take a look at our choices possible in the full house when we were looking at the possibilities of getting 3 of the 4 cards in any designation. You will notice that the order of the arrangement is rigid here, but there are many "permutations" or orders possible. The numerator of our fraction is actually giving us all the possible orders those cards might appear in. Our denominator allows us to reduce this figure to pick the number of combinations in any order.

If you go back to our first formula on the page you are probably wondering where that 40 came from. That is easy. Since straight flushes are also a species of the flush, they have to be taken out in order to come up with our chances of getting just a regular flush. You will see the same thing when we look at the straights. Now for the formula that gives us the actual odds of getting a flush:

5108/2,598,960 = 1 in 509

The numerator is the number of flushes possible, while the denominator is the number of all possible hands. The flush is one of the easiest hands to fill as the number of other cards available is pretty high. If you are dealt 4 of a suit and go for one card the chances of getting it are 9/47 (number of cards remaining in the suit over the number of unknown cards). This is nine times better than wishing for a single card out of the pack.

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