Basic Statistical Analysis of Poker Hands

This summer I've spent a lot of time playing poker. The game has both a psychological and mathematical aspect that I find very interesting. For the purpose of simplicity we will be sticking to the Texas Hold'em style of playing poker (in part because it is the most popular and in part because I am in Texas this summer).

How to Play

Texas Hold'em style poker passes out 2 cards to each player. Players can then look at their cards and bet. A first round of betting happens (referred to as 'pre-flop bets'). Once this occurs 3 community cards are placed on the table (the flop).

Betting takes place again, 1 more community card is placed, another round of betting, and the fifth final community card is placed (sometimes called the river). A final round of betting occurs and then players compare hands.

A player wins if they have the best 5-card hand they can make from the 7 cards visible to them (their 2 own and the 5 community cards).

Hand Rankings

The ranks of hands are the following. If two players have the same hand it goes to the player with the higher value. If the two values are equal then it goes to the 'kicker', the next highest card (which is dependent on the type of hand).

1. Royal Flush: Ace (A), King (K), Queen (Q), Jack (J), 10 of the same suite.
2. Straight Flush: 5 cards in a row of the same suite
3. Four of a Kind: 4 cards of the same rank, also referred to as "quads"
4. Full House: 3 cards of the same rank and 2 cards of the same rank
5. Flush: 5 cards of the same suite
6. Straight: 5 cards in a row
7. Three of a Kind: 3 cards of the same rank
8. Two Pair: 2 cards of the same rank and 2 other cards that are the same rank
9. Pair: 2 cards of the same rank
10. High Card: Single lone card

Examples

For example:

• An A high would lose to a pair of 2's, but an A high would win against a K high.
• A flush of Q,J,9,8,5 of hearts would lose to A,Q,8,5,2 of hearts (due to the A being higher than Q)

Basic Statistics

Source: [1]

The total number of distinct 7 hand cards are given by ${52 \choose 7} = 133,784,560$. The frequency of each card hand and associated percent chance of getting the hand (with lower hand excluding the higher hands, i.e. straight flush doesn't include royal flush in frequencies):

1. Royal Flush:

   $${4 \choose 1}{47 \choose 2} = 4,324$$

   with a resulting percentage of

   $$0.0032\%$$

2. Straight Flush:

   $${9 \choose 1}{4 \choose 1}{46 \choose 2} = 37,260$$

   with a resulting percentage of

   $$0.0279\%$$

3. Four of a Kind:

   $${13 \choose 1}{48 \choose 3} = 224,848$$

   with a resulting percentage of

   $$0.168\%$$

4. Full House:

   $${13 \choose 2}{4 \choose 3}^{2}{44 \choose 1} + {13 \choose 1}{12 \choose 2}{4 \choose 3}{4 \choose 2}^{2} + {13 \choose 1}{12 \choose 1}{11 \choose 2}{4 \choose 3}{4 \choose 2}{4 \choose 1}^{2} = 3,473,184$$

   with a resulting percentage of

   $$2.60\%$$

5. Flush:

   $${4 \choose 1} \times \left[{13 \choose 7} - 217\right] + {4 \choose 1} \times \left[\left[{13 \choose 6} - 71\right] \times 39\right] + {4 \choose 1} \times \left[\left[{13 \choose 5} - 10\right] \times {39 \choose 2}\right] = 4,047,644$$

   with a resulting percentage of

   $$3.03\%$$

6. Straight:

   $$217 \times \left[4^7 - 756 - 4 - 84\right] + 71 \times 36 \times 990 + 10 \times 5 \times 4 \times \left[256 - 3\right] + 10 \times {5 \choose 2} \times 2268 = 6,180,020$$

   with a resulting percentage of

   $$4.62\%$$

7. Three of a Kind:

   $$\left[{13 \choose 5} - 10\right]{5 \choose 1}{4 \choose 1}\left[{4 \choose 1}^4 - 3\right] = 6,461,620$$

   with a resulting percentage of

   $$4.83\%$$

8. Two Pair:

   $$\left[1277 \times 10 \times \left[6 \times 62 + 24 \times 63 + 6 \times 64\right]\right] + \left[{13 \choose 3}{4 \choose 2}^3{40 \choose 1}\right] = 31,433,400$$

   with a resulting percentage of

   $$23.5\%$$

9. One Pair:

   $$\left[{13 \choose 6} - 71\right] \times 6 \times 6 \times 990 = 58,627,800$$

   with a resulting percentage of

   $$43.8\%$$

10. High Card:

    $$1499 \times \left[4^7 - 756 - 4 - 84\right] = 23,294,460$$

    with a resulting percentage of

    $$17.4\%$$

This table summarizes the above.

Notice that having a one pair is the most likely, by almost twice that of having two pairs, and having JUST a high card is the third-most likely hand at $17.4\%$.