Poker Odds - Figuring Odds

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Figuring Odds - I have created this section to explain how I arrived at the odds of drawing poker hands. This example applies to 5 Card Stud, but these concepts can be applied to any game. I am not a mathematical genius, and you don't have to be either to understand the concepts below. These math formulas come out of an old basic statistics book and a pre-calculus textbook of mine. The skills used here can be applied to a wide range of calculating odds.

Factorials - A factorial means that you simply multiply the integers in a number. For example, for the number 4, you multiply 4x3x2x1=24. Imagine that you have 4 coffee cups. How many combinations can you arrange them in? The answer is 4!, or 24. There are obviously 4 positions to put the first cup , then there will be 3 positions left to put the second cup, 2 positions for the third cup, and only 1 for the fourth cup, or 4x3x2x1 = 24. If you had n cups there would be n(n-1)(n-2)* ... * 1 = n! ways to arrange them. Any scientific calculator should have a factorial button, usually denoted as x!, and the factor (x) function in Excel will give the factorial of x. (The total number of ways to arrange 52 cards would be 52! = 8.065818 x 1067.)

The Combinatorial Function - Now imagine that you have 10 coffee cups each of which is a different color. Imagine that you want to see how many different groups of 4 coffee cups out of the 10 coffee cups you could have. How many different combinations of coffee cups are there to choose from? The answer is 10! / (4!*(10-4)!) = 210. The general case is if you have to form groups of y coffee cups out of a total of x then there are x!/(y!*(x-y)!) combinations to choose from. Why? For the example given there would be 10! = 3,628,800 ways to put the 10 coffee cups in order. However you don't have to establish an order of the coffee cups or those that aren't in the group of 4. There are 4! = 24 ways to arrange the coffee cups in each grouping of 4 and 6! = 720 ways to arrange the other 6. By dividing 10! by the product of 4! and 6! you will divide out the order of coffee cups in and out of the total and be left with only the number of combinations, specifically (1*2*3*4*5*6*7*8*9*10)/((1*2*3*4)*(1*2*3*4*5*6)) = 210. The combination (x,y) function in Excel will tell you the number of ways you can arrange a group of y out of x.

Now we can determine the number of possible five card hands out of a 52 card deck. The answer is combine (52,5), or 52!/(5!*47!) = 2,598,960. If you're doing this by hand because your calculator doesn't have a factorial button and you don't have a copy of Excel, then realize that all the factors of 47! cancel out those in 52! leaving (52*51*50*49*48)/(1*2*3*4*5). The probability of forming any given hand is the number of ways it can be arranged divided by the total number of combinations of 2,598.960. Click here for the number of combinations for each hand. Just divide by 2,598,960 to get the odds.

Below are the odds of being dealt a specific hand using 5 cards.