Poker is not just a game of luck. It requires skill, strategy, and a good understanding of the probabilities involved.
One of the key concepts that players need to understand is combinatorics. In this article, we will delve into how combinatorics is used in poker.
Combinatorics is the study of counting and arranging objects. In poker, it refers to the number of possible ways certain hands can be made from a deck of cards. This knowledge can help players make better decisions about whether to bet or fold.
Let’s take an example: You are playing Texas Hold’em and have been dealt two cards – Ace of hearts and King of clubs. You want to know your chances of making a flush by the river if you decide to call.
To calculate this, you need to use combinatorics. There are 52 cards in a deck, and there are already two cards in your hand, which leaves 50 cards remaining. Of these 50 cards, 13 are hearts (including the Ace) and 9 have already been dealt (2 in your hand and 2 on the board), leaving 11 hearts remaining.
The number of ways you can make a flush by the river is calculated using combinations. A combination is a unique selection of items where order doesn’t matter. In this case, we want to select five cards from the remaining deck where three are hearts (to make our flush) and two are any other card.
The formula for calculating combinations is nCr = n! / r!
(n-r)!, where n represents the total number of items, r represents the number being chosen at once, and ! means factorial (multiply all integers up to that number).
So for our example:
n = 50 (remaining cards)
r = 5 (total number we want)
nCr = 50C5 = 2,118,760
This means there are 2,118,760 possible combinations of five cards that can be drawn from the remaining deck. But not all of these combinations will make a flush. We need to calculate the number of combinations where three cards are hearts and two are any other card.
The formula for calculating this is (n1Cr1) x (n2Cr2), where n1 represents the number of hearts remaining in the deck, r1 represents the number of hearts we need to draw, n2 represents the remaining cards that are not hearts, and r2 represents the number we need to draw.
n1 = 11 (remaining hearts)
r1 = 3 (hearts we need)
n2 = 39 (remaining non-hearts)
r2 = 2 (non-hearts we need)
(n1Cr1) x (n2Cr2) = 165 x 741 = 122,265
This means there are only 122,265 possible ways to make a flush by the river. To calculate the probability of making a flush, we divide this number by the total number of combinations:
Probability of making a flush = Number of ways to make a flush / Total number of combinations
Probability of making a flush = 122,265 / 2,118,760
Probability of making a flush = 0.058 or approximately 6%
So in this example, you have about a 6% chance of making a flush by the river if you decide to call.
Combinatorics can also be used in other aspects of poker such as calculating the probability of hitting certain hands on different streets or determining optimal bet sizing based on your hand and your opponent’s range.
In conclusion, combinatorics is an important concept in poker that can help players make better decisions and improve their overall game. By understanding how to calculate probabilities using combinatorics, players can make more informed decisions about when to bet, call, or fold.
