## Pocket Win Probabilities

Calculating the exact win probabilities of each pocket hand is getting close to impossible when too many opponents are sitting on the table (a game with 2 players has only around 3 billion possible combinations after the river, thats a number with 12 digits… a game with 16 players has 8 quindecillion possible combinations on the showdown, a number with 48 digits).

But an estimation is easy with a Monte Carlo Simulation: A program generates random hands for each player several million times, checks whom of them has the best hand and calculates the average winning rates until the numbers reach stable values… after each million hands the average is recalculated and compared to the last results, if the values do not change any more (except slightly somewhere behind the comma) the final result is reached.

The attached document shows win probabilities for each pocket hand on tables with 2-16 players (1 to 15 opponents).

Poker Cards (Monte Carlo Simulation)
(document updated on April 09 2016)

As in the last publications, the mathematical background is in German. But the relevant content (the numerical tables) can be used by everyone without problem, even if you are not speaking this language.

Note that these numbers are only valid if each player stays in the game until the showdown. Usually bad hands have an even lower win probability as they are often folded before the showdown (so they won’t win even if the best hand would come up in turn and river) which gives good hands a higher probability than shown in these tables…

## Probability to get a Hand

In case you wonder why some hands are more valuable than others:
The higher the probability to get a hand, the lower its value.

The probability to get a Royal Flush, a Straight Flush or a Four-of-a-Kind is less than one percent each (even after the river), thus they are the most valuable hands.
The probability to get a pair is already close to 50% on the flop, thus not very valuable.

The following document shows the probability to get each hand in flop, turn and river. For those curious enough, I added the mathematical formulas how to calculate these values.

Hand Probabilities

P.D: This document is still in German as I don’t see much value in translating the description of each mathematical formula. It is important to be published along with the data for traceability, but other than that it is irrelevant for any upcoming analysis.