We can calculate the probability of each type of hand of 5-card in poker.

### Basis of calculation and notations

The calculation of the probabilities of the various possible hands is mainly through the calculations of combinations. Remember that we note **( ^{n}_{p})** the number of combinations (without repetition) of

**p**elements from a set of

**n**elements.

However, note that for all n integer **( ^{n}_{1})** =

**(**=

^{n}_{n-1})**n**. In fact the number of possible choices of an element of

**n**is simply… equal to

**n**. And choosing

**(n-1)**means choosing the one that deviates. So in this case as there are also

**n**possibilities.

In what follows we note **N** the number of values. So the total number of cards is **4N**.

And we note **S** the number of accepted straights.

- For a deck of 52 cards, there are
**N = 13**and counting the straights of A-2-3-4-5 (white straight) to 10-J-Q-K-A (royal flush), is obtained**S = 10**. If the white straight is not allowed, one has only**S = 9**. - For a set of 32 cards, there are
**N = 8**and straights from 7-8-9-10-V to 10-J-Q-K-A, and is obtained**S = 4**.

### Draw poker: Basic hands

Each player receives five cards systematically. They are all private and closed.

The total number of combinations of 5 cards among 4N of the game is **( ^{4n}_{5})**.

#### Table of probabilities of each hand

This first table shows the odds of each hand for games of 52 and 32 cards. In the deck of 52 cards, calculations include “extended straights”, that is to say that the combination 2-3-4-5-A (straight white) is considered a straight. Therefore S = 10 as described above.

Following the practice, the hands of this table are mutually exclusive:

- for the
*straight flushes*count only non-royal flushes, - the
*flushes*are those that are not straights, - the
*straights*are those that are not flush or royal flush, - etc.

Hand | Formula | 52 cards (N=13 and S=10) | 32 cards (N=8 and S=4) | ||
---|---|---|---|---|---|

combinations | combinations | probability | combinations | probability | |

Royal flush | 4 | 4 | 0,000154 % | 4 | 0,00199 % (1/50344) |

Straight flush | 4(S-1) | 36 | 0,00139 % | 12 | 0,0060 % (1/16781) |

Four of a kind | 4N(N-1) | 624 | 0,024 % | 224 | 0,111 % (1/899) |

Full house | 24N(N-1) | 3 744 | 0,144 % | 1 344 | 0,667 % (6/899) |

Flush | 4 ( (^{N}_{5}) − S ) |
5 108 | 0,197 % | 208 | 0,103 % |

Straight | 1020 S | 10 200 | 0,392 % | 4 080 | 2,026 % |

Three of a kind | 32 N(N-1)(N-2) | 54 912 | 2,113 % | 10 752 | 5,339 % |

Two pair | 72 N(N-1)(N-2) | 123 552 | 4,754 % | 24 192 | 12,013 % |

One pair | 64 N(N-1)(N-2)(N-3) | 1 098 240 | 42,257 % | 107 520 | 53,393 % |

High card | 1020 ( (^{N}_{5}) − S ) |
1 302 540 | 50,118 % | 53 040 | 26,339 % |

Total | (^{4N}_{5}) |
2 598 960 | 100 % | 201 376 | 100 % |

We notice:

- that in the case of the game of 32 cards the order of probability of hands does not match the order of their strength: the color is more rare than the square and high card rarer than a pair.
- that hands “served” over the three of a kind are extremely rare: less than 1% of hands in 52 cards, and less than 3% to 32 cards.

#### Probability table to have at least …

In practice, the vast majority of games are played in the lower zone: high card, one pair, playable drawing, two pair or three of a kind. These are the hands that must be studied to discuss the risks of openings and revival levels.

Among the hands “high card”, it says there is a **draw** in the special cases where we need to exchange (*draw*) one card to form a color (*flush draw*) or a straight (*straight draw*).

The *draw* is considered superior to *one pair*. This means that generally, after exchange of a card, the resulting hand is often better than that which is obtained by improving a pair.

If we talk about the types of draws, *flush draw* is now a little weak, and could be downgraded in the following table.

In this table, the probabilities are independent of the number of players. In addition, the white straights are not taken into account (but without significant impact on the figures).

Reading is direct: for example, with a deck of 48-cards, the probability that a player is better than a pair of aces is 16.1%.

Number of cards in deck | |||||
---|---|---|---|---|---|

Hand | 52 | 48 | 44 | 40 | 32 |

Three of a kind | 0,7 % | 0,9 % | 1,1 % | 1,5 % | 2,9 % |

Two pair | 2,8 % | 3,3 % | 4,1 % | 5,1 % | 8,8 % |

Draw | 7,5 % | 8,9 % | 10,7 % | 13,0 % | 20,8 % |

Pair As | 14,4 % | 16,1 % | 18,4 % | 21,5 % | 30,9 % |

Pait King | 17,5 % | 19,6 % | 22,4 % | 26,0 % | 36,8 % |

Pair Queen | 20,5 % | 23,1 % | 26,4 % | 30,5 % | 42,7 % |

Pair jack | 23,6 % | 26,6 % | 30,3 % | 35,0 % | 48,6 % |

Pair 10 | 26,7 % | 30,1 % | 34,3 % | 39,5 % | 54,5 % |

Pair 9 | 29,8 % | 33,6 % | 38,2 % | 44,0 % | 60,4 % |

Pair 8 | 32,8 % | 37,0 % | 42,2 % | 48,6 % | 66,3 % |

Pair 7 | 35,9 % | 40,5 % | 46,1 % | 53,1 % | 72,2 % |

Pair 6 | 39,0 % | 44,0 % | 50,1 % | 57,6 % | |

Pair 5 | 42,1 % | 47,5 % | 54,1 % | 62,1 % | |

Pair 4 | 45,2 % | 51,0 % | 58,0 % | ||

Pair 3 | 48,2 % | 54,4 % | |||

Pair 2 | 51,3 % |

The table can answer questions like:

I have a pair of king served, we play four with 32 cards, what is the probability that my hand is the best?

Here is the calculation steps in response:

- The probability for a player to have at least a pair of kings is: 36.8%. This is the probability of having more than a pair of queens. It therefore has 63.2% chance of having strictly less than a pair of kings.
- For the pair of kings to be the strongest, the first opponent must have less AND the second to have less and the third to have less.

The probability of this event is the product: 63.2% x 63.2% x 63.2% = 25.24%

- Our pair of kings therefore has 25.24% chance of beating the other three players.

## Keith Wilson

It gets a bit more challenging with my deck especially when you bring in rainbow hands. Grappling with a php script for the Draw page.