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seven card poker odds, Andrew Hagenbush and Mike Ulasewicz
7- Card Poker Hand Odds
By Andrew Hagenbush (A1)
and Featuring Michael Ulasewicz(A1)
This tutorial will go through the odds of getting several different poker hands when dealt 7 cards. If you are an IE this will help you in most of your ISyE classes because combinations and probabilities are incorporated in much of the material you will cover. Either way it will increase your understanding of how the game of poker is played and maybe save you from losing your money in Vegas!
In the following examples we will be using a standard deck that has 52 cards. Each card has a unique rank (a number between 2 and 10 and Jack, Queen, King, and Ace) and suit (Clubs, Hearts, Spades, or Diamonds) This equals 13 ranks and 4 suits.
We will represent these suits and ranks as follows
Suits: c, s, d, h
Ranks: A 2 3 4 5 6 7 8 9 10 J Q K
For example the Ace of Hearts would be Ah.
For each example, the probabilities will be calculated without replacement and without considering order.
In order to understand how these probabilities are calculated, we must briefly explain how combinations work.
To find the number of combinations when not considering order we will use this formula.
The notation for choosing r elements from a group of n elements:
nCr = n! / (k! x (n-k)!)
The left side's notation is read as "n choose k" and be refered to as
such throughout the following examples.
Before we begin showing you a few example probabilities, first we need to briefly explain what happens when we compute these probabilities. First, all probabilities are between 0 and 1, with 1 being the probability of something that is 100% sure of happening. The following probabilites are all computed as the number of times the event could happen divded by the total number of events that exist in the sample space, thus always giving us a number between 0 and 1.
Now we will go through our examples to show the odds of getting certain poker hands.
The hands we will evaluate are, four of a kind and three random cards, three of a kind three of another kind and one random card, 7-card flush, and a four of a kind and three of another kind. By evaluating these examples we will demonstrate the proper way to find the probability of any number of poker hands using counting.
Example 1:
Four of one kind, and three random cards (commonly expressed as XXXXYZW):
(13C1) x (4C4) x (12C3) x (4C1)^3
/
(52C7)
.001368
Important Explanation:
To put this into words, we are choosing 1 rank out of the 13 total ranks, then out of the 4 suited cards in that rank, we are choosing all 4 of those cards. Next, out of the remaining 12 ranks (we already chose one, so we cannot choose it again), we choose 3 of them. Out of the 4 suits of each of these ranks, we choose 1 from each of the 3 cards. Finally, we divide this product by the total number of 7 card hands that can be chosen from a 52 card deck (52 choose 7).
Example 2:
Three of one kind, three of another kind, and one random card (expressed as XXXYYYZ):
(13C2) x (4C3)^2 x (11C1) x (4C1)
/
(52C7)
.00041
Example 3:
A seven card flush (all cards in the same suit):
(4C1) x (13C7)
/
(52C7)
.000051
Important Explanation:
This is slightly different than the above problems. First we choose one suit out of the 4 suits. Next, we choose 7 cards out of the 13 ranks in that suit and multiplying them together. Finally, we divide that product by the total number of ways to choose a seven card hand.
Example 4:
Four of one kind and three of another (expressed as XXXXYYY):
(13C1) x (4C4) x (12C1) x (4C3)
/
(52C7)
.000005
These examples should help you to understand how to solve problems like this in the future.
by using the examples shown above anyone with a basic understanding of combinations should be able to calculate the odds of getting any number of different card combinations. many times students find themselves wondering how the math skills they learn in school can help them in everyday life. well, we have shown in the above that if you are part of the growing number of people interested in playing the game of poker you can benefit greatly from the knowledge of combinations and probability. hopefully this information will help you understand poker, combinations, and maybe even save you from gambling away your tuition money!
love,
Michael Ulasewicz
Andrew Hagenbus
