Poker

In the world of poker, one of the key skills is knowing the probabilities. Players who have a solid foundation in mathematical probabilities, have an advantage that can move them up the hierarchy of successful players. Knowledge of probabilities allows them to better estimate what their chances of improving their hand, making a stronger hand or, conversely, what the risks are of continuing to play. This allows them to play strategically and more wisely, which is crucial for long-term success at the poker table. Regardless of a player's experience and level, knowing probabilities is a valuable tool for anyone looking to improve their poker skills.

In this article, we will discuss the probabilities in the most famous variant of Texas holdem.

Probability
Table

Probability

The first thing to consider is how many possible combinations of cards there are for a player in poker. In total, we are playing with a deck of 52 cards (4 x 13), always picking 7 cards (2 player + 5 table).

That makes 133,784,560 combinations.

\begin{aligned} \binom{52}{7} = \frac{52!}{7!(52-7)!} = 133\;784\;560 \end{aligned}

Royal Flush

A Royal Flush is the rarest and most powerful hand in poker. At the same time, calculating the probability of getting it is relatively simple.

A Royal Flush can be made in only 4 ways:

10♠-J♠-Q♠-K♠-A♠
10♣-J♣-Q♣-K♣-A♣
10♥-J♥-Q♥-K♥-A♥
10♦-J♦-Q♦-K♦-A♦

If we played with only 5 cards in total, the calculation would be easier. Because we would know that there are only 4 possible combinations of all the cards.

However, we are playing with a total of 7 cards, so we must also include the remaining 2 cards in our calculation of combinations.

\begin{aligned} 4 * \binom{47}{2} = 4 * 1\;081 = 4\;324 \end{aligned}

Binomial coefficient 47 choose 2 because 47 (52 - 5, the remaining cards in the deck) and 2 (the last 2 cards that can be placed on the table).

Out of 133,784,560 combinations, only 4 324 can be hit for a royal flush.

\begin{aligned} 4\;324 : 133\;784\;560 = 1 : 30\;940 ≈ 0.00323 \% \end{aligned}

The probability of a royal flush is approximately 0.00323%.

Straight flush

Unlike the royal flush, there are many more ways to deal, as we are not limited by the mandatory 10 to A sequence. While this increases the probability, it also makes our calculation more difficult.

How many ways are there?

A-2-3-4-5
2-3-4-5-6
3-4-5-6-7
4-5-6-7-8
5-6-7-8-9
6-7-8-9-10
7-8-9-10-J
8-9-10-J-Q
9-10-J-Q-K
~~10-J-Q-K-A~~

If we don't count the royal straight, we now have 36 (9 * 4) ways.

\begin{aligned} 36 * \binom{46}{2} = 36 * 1\;035 = 37\;260 \end{aligned}

This time we are not using 47 in theBinomial coefficient, but 46. This is because we are also not counting the highest A card, which is only used for the royal flush.

Of the 133,784,560 combinations, only 37 260 can be hit for a straight flush (41 584 including the royal flush).

\begin{aligned} 37\;260 : 133\;784\;560 ≈ 1 : 3\;590.57 ≈ 0.02785 \% \end{aligned}

The probability of a straight flush is approximately 0.02785% (including a royal flush approximately 0.03105%).

Poker / Four of a kind

This time we are picking four of each rank card. We have 13 ways to do that.

\begin{aligned} 13 * \binom{48}{3} = 13 * 17\;296 = 224\;848 \end{aligned}

Of the 133,784,560 combinations, only 224 848 can be hit for a poker.

\begin{aligned} 224\;848 : 133\;784\;560 = 1 : 595 ≈ 0.16807 \% \end{aligned}

The probability of a poker is approximately 0.16807%.

Full house

Full house is a combination of 1 three of a kind and 1 pair. In terms of combinatorics it can be obtained in 3 ways. We calculate all 3 and sum them up at the end.

First way

The first way can be 2 triples and 1 remaining card.

\begin{aligned} \binom{13}{2} * \binom{4}{3}^2 * \binom{44}{1} = 54\;912 \end{aligned}

Second way

The second way can be 1 triple and 2 pairs.

\begin{aligned} \binom{13}{1} * \binom{4}{3} * \binom{12}{2} * \binom{4}{2}^2 = 123\;552 \end{aligned}

A triple can be made up of 13 different card ranks (A,2,3,...K), each of these triples can be made up in 4 suits (♠-♣-♥, ♠-♣-♦, ♠-♥-♦, ♣-♥-♦). Pairs can be composed of the remaining 12 card ranks, each pair 6 (squared, since we are counting 2 pairs) in suit ways.

Third way

The third and most probable way can be 1 triple, 1 pair and 2 remaining unpaired cards.

\begin{aligned} \binom{13}{1} * \binom{4}{3} * \binom{12}{1} * \binom{4}{2} * \binom{11}{2} * \binom{4}{1}^2 = 3\;294\;720 \end{aligned}

A triple can be made up of 13 different card ranks, in 4 suit ways. A pair can be made up of the remaining 12 card ranks, 6 suit ways. The last 2 unpaired cards can be made up of 11 card ranks, 4 (squared, since we are counting 2 cards) suit ways.

Final probability

Sum up all the ways mentioned above.

\begin{aligned} 54\;912 + 123\;552 + 3\;294\;720 = 3\;473\;184 \end{aligned}

Of the 133,784,560 combinations, only 3 473 184 can be hit for a full house.

\begin{aligned} 3\;473\;184 : 133\;784\;560 ≈ 1 : 38.52 ≈ 2.5961 \% \end{aligned}

The probability of a full house is approximately 2.5961 %.

Flush

We will also count the combinations to get a flush in 3 ways. Each calculation will have one thing in common and that will be the number 4 (suit). In the calculations, we will subtract the straight flushes and the royal flushes.

First way

The first case is a hand where all 7 cards are of the same suit.

\begin{aligned} 4 * \Bigg[ \binom{13}{7} - 217 \Bigg] = 5\;996 \end{aligned}

Second way

The second way is a combination of 6 cards of the same suit and 1 card of a different suit.

\begin{aligned} 4 * \Bigg[ \binom{13}{6} - 71 \Bigg] * 39 = 256\;620 \end{aligned}

Third way

The third way is a combination of 5 cards of the same suit and 2 cards of a different suit.

\begin{aligned} 4 * \Bigg[ \binom{13}{5} - 10 \Bigg] * \binom{39}{2} = 3\;785\;028 \end{aligned}

Final probability

Sum up all the ways mentioned above.

\begin{aligned} 5\;996 + 256\;620 + 3\;785\;028 = 4\;047\;644 \end{aligned}

Of the 133,784,560 combinations, only 4 047 644 can be hit for a flush.

\begin{aligned} 4\;047\;644 : 133\;784\;560 ≈ 1 : 33.05 ≈ 3.02549 \% \end{aligned}

The probability of a flush is approximately 3.02549% (including a straight flush and a royal flush approximately 3.05654%).

Straight

Again, we split the calculation into 3 ways. In the calculations we will subtract straight flushes and royal flushes.

First way

7 different card ranks (for example, 4-5-6-7-8-J-K).

\begin{aligned} 217 * (4^7 - 756 - 4 - 84) = 3\;372\;180 \end{aligned}

Second way

6 different card ranks (for example, 4-5-6-7-8-8-K).

\begin{aligned} 71 * 36 * 990 = 2\;530\;440 \end{aligned}

Third way

5 different card ranks (for example, 4-5-6-7-8-8-8).

\begin{aligned} 10 * 5 * 4 * (256 - 3) + 10 * \binom{5}{2} * 2268 = 277\;400 \end{aligned}

Final probability

Sum up all the ways mentioned above.

\begin{aligned} 3\;372\;180 + 2\;530\;440 + 277\;400 = 6\;180\;020 \end{aligned}

Of the 133,784,560 combinations, only 6 180 020 can be hit for a straight.

\begin{aligned} 6\;180\;020 : 133\;784\;560 ≈ 1 : 21.65 ≈ 4.61938 \% \end{aligned}

The probability of a straight is approximately 4.61938% (including a straight flush and a royal flush approximately 4.65043%).

Three of a kind

Each triple must consist of 3 same card ranks and 4 different card ranks (for example, A-A-A-4-A-4-5-6-7).

\begin{aligned} \Bigg[ \binom{13}{5} - 10 \Bigg] * 5 * 4 * (4^4 - 3) = 6\;461\;620 \end{aligned}

Each three of a kind must consist of 5 five different card ranks (to avoid, for example, a full house), and at the same time we must subtract the 10 possibilities that make up the straight. There are 5 possibilities for the order of the three of a kind and 4 suit ways.

The remaining 4 cards can be of any suit, but all 4 cannot be the same suit as one of the cards in the triple.

Of the 133,784,560 combinations, only 6 461 620 can be hit for a three of a kind.

\begin{aligned} 6\;461\;620 : 133\;784\;560 ≈ 1 : 20.7 ≈ 4.82987 \% \end{aligned}

The probability of a three of a kind is approximately 4.82987%.

Two pair

A two pair hand also consists of 5 different card ranks (for example, K-K-J-J-4-5-6). Paradoxically, the probability on two pair is higher than on no pair. We will divide the calculation in 2 ways.

First way

3 pairs and 1 different card.

\begin{aligned} \Bigg[ \binom{13}{5} - 10 \Bigg] * 10 * (6 * 62 + 24 * 63 + 6 * 64) = 28\;962\;360 \end{aligned}

Second way

2 pairs and 3 different cards.

\begin{aligned} \binom{13}{3} * \binom{4}{2}^3 * \binom{40}{1} = 2\;471\;040 \end{aligned}

Final probability

Sum up all the ways mentioned above.

\begin{aligned} 28\;962\;360 + 2\;471\;040= 31\;433\;400 \end{aligned}

Of the 133,784,560 combinations, only 31 433 400 can be hit for a two pair.

\begin{aligned} 31\;433\;400 : 133\;784\;560 ≈ 1 : 4.26 ≈ 23.49554 \% \end{aligned}

The probability of a two pair is approximately 23.49554%.

Pair

A pair consists of 6 different card values (for example, J-J-2-3-6-7-8).

\begin{aligned} \Bigg[ \binom{13}{6} - 71 \Bigg] * 6 * 6 * 990 = 58\;627\;800 \end{aligned}

Of the 133,784,560 combinations, only 58 627 800 can be hit for a pair.

\begin{aligned} 58\;627\;800 : 133\;784\;560 ≈ 1 : 2.28 ≈ 43.82255 \% \end{aligned}

The probability of a pair is approximately 43.82255%.

No pair / high card

As a last possibility, in poker, you can get no hand. In this case, you can just rely on the rank of your highest card.

As mentioned, paradoxically there is a better chance of getting one or even two pairs than no pair.

\begin{aligned} 1499 * (4^7 - 756 - 4 - 84) = 23\;294\;460 \end{aligned}

Of the 133,784,560 combinations, only 23 294 460 can be hit for no pair.

\begin{aligned} 23\;294\;460 : 133\;784\;560 ≈ 1 : 5.74 ≈ 17.41192 \% \end{aligned}

The probability of no pair is approximately 17.41192%.

Table

Hand	Probability	Odds
Royal flush	0.00323%	1 : 30 939
Straight flush	0.02785%	1 : 3589.57
Poker / Four of a Kind	0.16807%	1 : 594
Full house	2.5961%	1 : 37.52
Flush	3.02549%	1 : 32.05
Straight	4.61938%	1 : 20.65
Three of a kind	4.82987%	1 : 19.7
Two pair	23.49554%	1 : 3.26
Pair	43.82255%	1 : 1.28
No pair / high card	17.41192%	1 : 5.74
Total	100%	1 : 0