(Poker continued)
Poker Probability
What are the chances, before the deal, that a player will end up with any particular poker hand?
Of course, the number of cards drawn from which the 5 cards are chosen is an important factor in determining this. In games with community cards, the community cards will not increase the likelihood of any specific player drawing a particular hand. Community cards will, however, increase the likelihood that a particular hand is made amongst all the players, since all the players are playing most of the same cards.
The following charts cover pre-deal poker probabilities for 5 card games and 7 card games, without draws or wild cards. Following the charts is a breakdown of how I calculated the poker probabilities for the 5 card game. The calculations for the 7 card game are a little more complex; so I wimped out and stole them from wikipedia (i). (The wiki article combined the straight and royal flush calculations, but I split them up.)
Because of the number of possible combinations after the hand is dealt, it would be rather pointless to attempt to list the probabilities for any given situations. Of course, the decisions you make in poker are largely influenced on the probability of having the best hand. I will cover some of this later.
| Pre-draw probabilities for 5 card games |
| Hand | Frequency | Probability | Odds |
| Royal Flush | 4 | .000001539 | 649,739 : 1 |
| Straight Flush | 36 | .00001385 | 72192.3 : 1 |
| 4 of a kind | 624 | .0002401 | 4,164 : 1 |
| Full House | 3,744 | .001441 | 693.17 : 1 |
| Flush | 5,108 | .001965 | 507.80 : 1 |
| Straight | 10,200 | .003925 | 253.8 : 1 |
| 3 of a kind | 54,912 | .02113 | 46.36 : 1 |
| Two pair | 123,552 | .04754 | 20.04 : 1 |
| Pair | 1,098,240 | .4226 | 1.37 : 1 |
| High card | 1,302,540 | .5012 | .995 : 1 |
| Pre-draw probabilities 7 card games (i). |
| Hand | Frequency | Probability | Odds |
| Royal Flush | 4,324 | .0000323 | 30,939 : 1 |
| Straight Flush | 37,260 | .0002785 | 3,590 : 1 |
| 4 of a kind | 224,848 | .00168 | 594 : 1 |
| Full House | 3,473,184 | .0260 | 37.5 : 1 |
| Flush | 4,047,644 | .0303 | 32.1 : 1 |
| Straight | 6,180,020 | .0462 | 20.6 : 1 |
| 3 of a kind | 6,461,620 | .0483 | 19.7 : 1 |
| Two pair | 31,433,400 | .235 | 3.26 : 1 |
| Pair | 58,627,800 | .438 | 1.28 : 1 |
| High card | 23,294,460 | .174 | 4.74 : 1 |
Poker Probability Methods
The following is not meant as a thorough lecture on poker probability (not
that I would be qualified to deliver such a lecture) but more to give the reader
insight into my methods. The general method I use to determine the probability of any given poker hand is as follows:
- Organize the hand into appropriate subsets
- Determine the number of combinations of ranks, avoiding overlaps *.
- Determine the number of combinations of suits, avoiding overlaps *.
- Multiply number combination of suits by number combinations of ranks, giving the number of possible
combinations of the hand
- Divide the result from # 4 by the number of possible unique hands for the particular poker game.
* Avoiding overlaps requires some thought. For example, when calculating the
number of straight flushes, make sure you are not also counting royal flushes.
Remember, poker has 13 ranks (2-Ace) and 4 suits.
FORMULA # 1: To determine the number of combinations of n objects taken r at a time,
a formula is used which is denoted in a few different ways:
I prefer the later which reads "n choose r". The formula is n!/r!(n-r)! ("!" means factorial - the factorial of 5 would be 5 * 4 * 3 * 2 * 1).
FORMULA #2: Also, the way to determine the number of combinations of n objects taken r at a time in a sequence (meaning n, n+1 ,n+2...n+m) turns
out to be n - r + 1 (use for straights). Keep in mind that aces can be high or low (this will often make n = 14).
Five Cards
Total number of possible unique hands in 5 cards: 52!/5!(52-5)! = 2,598,960
Royal Flush
- 1 subset for the rank and 1 subset for the suit.
- Using formula #2, n = 5 because we have a total of 5 cards to choose from (Ace, King, Queen, Jack, Ten).
r = 5 because this is the number of cards in the straight. The number of possible rank combinations =
5-5 + 1 = 1.
- We can see that there are 4 possible suits, but for the sake of completeness
4C1 = 4.
- Multiply 4 * 1 = 4
- 4/2,598,960 = .000001539
Straight Flush
- 1 subset for the rank and 1 subset for the suit.
- Using formula #2, n = 13(Ace,2,3,4,5,6,7,8,9,10,J,Q,K) (n does not equal 14 here, because the Ace high straight flushes are
considered royal flushes). r = 5 (number in the sequence), so 13 - 5 + 1 = 9 which is the
number of possible rank combinations.
- The number of possible suit combinations: 4C1 = 4.
- 9 * 4 = 36
- 36/2,598,960 = .00001385
4 of a Kind
- 2 subsets for the rank and 2 subsets for the suits
- The ranks are divided into two subsets. Using formula # 1, the first subset =
13C1 = 13, because we have 13 cards to choose one possible rank. The second subset can
be made up of any of the remaining 12 cards, so 12C1 = 12. Multiplying
these two together will give you the number of possible rank combinations = 156.
- The number of suit subsets = 2. The first subset is 4C4 (4 suits, choose all) = 1. The second
subset = 4C1 = 4. 4 * 1 =4.
- 4 * 156 = 624
- 624/2,598,960 = .0002401
Full House
- 2 subsets for the rank and 2 subsets for the suits
- For the ranks, the first subset has 1 rank, so 13C1 = 13. The second subset can be any
of the other 12 ranks so 12C1 = 12. 12 * 13 = 156.
- For the suits, the first subset = 4C3 = 4 and the second = 4C2 = 6. 4 * 6 = 24.
- 24 * 156 = 3,744
- 3,744/2,598,960 = .001441
Flush
- 1 subset for the rank and 1 subset for the suits
- 13C5 - 14 - 5 + 1= 1,277 possible rank combinations (here 14 - 5 + 1 is the number of straight flushes including
royal flushes).
- Number of suits 4C1 = 4
- 1,277 * 4 = 5,108
- 5,108/2,598,960 = .001965
Straight
- 1 subset for the rank and 5 subset for the suits
- Formula # 2 : 14 - 5 + 1 = 10
- 5 subsets of 4C1 = 4^5 subtracting the number of ways they can all be the same
suit - 4 (4C1 * 1C1 * 1C1 * 1C1 * 1C1) = 1,020.
- 10 * 1,020 = 10,200
- 10,200/2,598,960 = .003925
Three of a Kind
- 2 subsets for the ranks and 3 subsets for the suits
- Ranks 13C1 and 12C2 = 66 * 13 = 858
- Suits 4C3 * 4C1 * 4C1 = 64
- 858 * 64 = 54,912
- 54,912/2,598,960= .02113
Two Pair
- 2 subsets for the rank and 3 subsets for the suits.
- Ranks 13C2 and 11C1 = 78 * 11 = 858
- Suits 4C2 * 4C2 * 4C1 = 144
- 858 * 144 = 123,552
- 123,552/2,598,960 =.04754
Pair
- 2 subsets for the rank and 4 subsets for suits.
- Ranks 13C1 * 12C3 = 13 * 220 = 2,860
- Suits 4C2 * 4C1 * 4C1 * 4C1 = 384
- 2,860 * 384 = 1,098,240
- 1,098,240/2,598,960 =.4226
Nothing
- 1 subset for rank and 5 subsets for suits.
- Rank 13C5 (all must be dif ranks)= 1,287 - 14 - 5 + 1 (subtract straights) = 1,277
- Suit 4^5 - 4 (subtract flushes) = 1,020
- 1,020 * 1,277 = 1,302,540
- 1,302,540/2,598,960 = .5012
Reference
- Wikipedia: Poker Probability
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