Poker Math

For a Five Card Stud, here are some variations

I’ve had so many questions about calculating the odds of drawing each poker hand that I’ve decided to write a section on the topic. Mathematical skill is assumed; anyone who has taken high school math should be able to follow this explanation. A wide range of probability problems can benefit from the techniques demonstrated here.

The Function of Factorials

Those familiar with the factorial function can skip this section. Keep reading if you think 5! means calling out the number five.

It’s likely that the manufacturer’s instructions for your couch call for frequent cushion rearranging. If your couch has four cushions, we’ll go with that. Is there any limit to the number of ways you may arrange them? There are four of them! There are four places to put the first cushion, three areas for the second, two for the third, and one for the last, or 4*3*2*1 = 24 places to put the pillows in. It’s possible to arrange n! cushions in various ways if you have n cushions. The fact(x) function in Excel provides the factorial of x, like any scientific calculator with a factorial button, typically written as x!. If you had 52 cards, the total number of possible ways to arrange them would be 52!

The Function of Combinatorial

To construct a committee of four persons, you’ll need to select four people from the pool of ten in your office. There must be an infinite number of possible combinations of individuals. 10!/(4!*(10-4)!) = 210 is the answer. There are x!/(y!*(x-y)!) combinations if you have to construct a committee of y people from a pool of x. Why? 10! = 3,628,800 different ways to arrange your office staff. The committee consists of the first four people listed, while the remaining six are considered lucky. However, you do not need to establish a hierarchy between the members of the committee and those who are not. People can be arranged in 4! = 24 different ways on the committee; 6! = 720 different arrangements are possible on the other two. To get the number of combinations, divide 10! by the product of 4! and 6! You’ll get 210, which is how many people are in and out of the committee. There are many ways to organize a group of y from x using Excel’s combine(x,y).

The number of viable five-card hands from a 52-card deck is now in our hands. In this case, the answer is combin(52,5), or 2,598,960. If you’re doing this manually because your calculator doesn’t have a factorial button and you don’t have a copy of Excel, be aware that all 47! Those in 52 are gone! In this case, we are left with (52 * 51 * 50 * 49 * 48). Any specific hand’s probability is calculated by dividing its possible configurations by 2,598.960, the total number of possible configurations. For convenience, we’ve listed the number of possible combinations for each hand below. To calculate the likelihood, multiply by 2,598,960.