Mathematics and probability of Hands in Poker
Whatever poker game is discussed, there are strictly defined names of poker hands. There are only nine of them, but more and more poker games appear on their basis.
In this article, we will tell you about the method of counting the number of possible combinations in poker, and how do it itself. At the end of the article there is a table showing the quantity and probability of combination formation. I have to say that here the word “poker” refers to a five-card oasis poker for a standard deck of 52 cards.
In order to be able to assess probabilities in poker tasks, you should know the basics of combinatorics and probability theory. In the meantime let’s go without them and count how many Royal Flush combinations exist (five cards of the same suit from Ten to Ace). I hope no one will argue with the fact that there are only four of them, one in each suit.
Also, without formulas, just using your brain, you can count the number of Four of a Kind combinations (four cards of the same rank).
There are 13 ranks of cards in the deck – from Two to Ace. Accordingly, Four of a Kind may have 13 values. A combination (hand) consists of five cards. So, in each of the 13 Fours of a Kind of different ranks, the fifth card may be any of the remaining cards in the deck. The number of cards in the deck is 52, we already use four of them. It remains 48 free cards that we can add one by one to four cards of one rank to get a new Four of a Kind combination each time. Multiplying 13 by 48, we get 624 – the number of possible different 5-card Four of a Kind combinations.
Thus, without complicated calculations we already have two numbers – 4 and 624. Looking at them, we can conclude that Royal Flush is 156 times higher than Four of a Kind (after all, it is dealt 156 times rarer). Standard poker payouts are 20:1 BET for Four of a Kind and 100:1 for Royal Flush. So why such a strong combination of Royal Flush is all paid just five times more than Four of a Kind? It turns out that the developers of the rules did not take into account probabilities, or may be the case is other? The answer is simple: mismatch of the level of cash payments for combinations against their actual mathematical value is just a conventionality making the game more interesting.
The nicer (stronger) poker hand, the easier it is to calculate the probability of its dealing. According to such logic, it turns out that the most difficult task is to count the number of five-card hands containing a pair. Yes, to do this, you have to use special formulas of combinatorics. But to count the number of empty hands in poker (“no game”) is even more difficult.
Now, the last time I suggest simply counting the number of Straight Flush combinations, and then referring to the formulas, and calculation of the remaining combinations will be faster.
The lowest Straight Flush ends by Five. The highest – by King. Total: the number of Straight Flush combinations of a certain suit is 9. By multiplying nine by four suits we have 36 – the number of Straight Flush combinations.
We will need the numbers 4, 36, 624 for further calculations. And now I want to introduce you very powerful and at the same time easy to remember formulas of combinatorics for almost all occasions.
Cn = n!
Recall that factorial is an operation in which the number n subjected to it is multiplied by (n-1), then multiplied by (n-2) and so on down to one.
Example 3! = 3*2*1 = 6 Practical application: Three people of different height can be lined up in a row six different ways. Or: you can eat a three-course meal six different ways, changing the sequence.
Formula two and last. The number of combinations of n by m. You can record it as Cnm
Cnm = n!/m!*(n-m)!
This formula is not just ideal for calculating the number of combinations in poker, but is also useful for other practical problems. You just need to understand how and in what conditions to use it.
The next combination after Four of a Kind (in terms of ease of counting) is Flush. Finding the number of Flush combinations will be a good example of the application of the formula for the number of combinations of n by m. We need to calculate how many different 5-card Flush combinations can be composed of 13 cards of the same suit? Let’s use the formula. The number n here is 13 – the number of cards of the same suit, and the number m is 5 – the number of cards we need to make Flush.
Cnm = C135 = 13!/5!*(13-5)! = 1287
1287 five-card combinations of a separate suit. By multiplying by four (by the number of suits), we get 5148. Now, subtracting 4 Royal Flushes and 36 Straight Flushes from this number, we obtain the correct number of 5108 - the number of different Flush five-card hands.
Totally we have:
Royal Flush 4
We have missed Full House, let’s count. This poker hand consists of three cards of one rank and two cards of another rank. Let’s take four cards of the same rank in hands. By folding one card, you can collect three cards of one rank in four ways.
Let’s check, using the same formula Cnm
And you can collect two pair cards of four cards of one rank in six ways:
C42=4!/2!*(4-2)!=6 the number of options of pair cards of one rank
Considering the particular example, let’s take three Aces. To collect Full House, we should add any pair to these cards. Total, the deck comprises 6*12 pairs (12 is the number of free ranks from 2 to King). 6 is the number of options of pair cards of one rank. Total: 4*6*12=288 possible Full Houses including three Aces. Where 4 is the number of ways to collect three cards of one rank. By multiplying this number by the number of ranks 13, we get 3744 - the number of Full Houses in poker deck.
Next, let’s continue with the most popular poker combination – Straight. It is truly popular. The existence of a variety of poker terms associated with Straight or its possible dealing proves it. I mean Open-ended Straight, Gutshot Straight, and other poker concepts.
In order to move from simple to complex concepts, consider a combination of Straight Flush from 2 to 6. Now change Two to Two of another suit, making a simple Straight of Straight Flush. In order to get more and more Straights we just have to change the suits of Straight-forming cards, leaving their rank unchanged. In this case, we can sequentially change suits of five cards – Twos, Threes, Fours, Fives and Sixes:
By changing the suit of one of these cards we get a NEW Straight.
Resorting to such sorting, we can accurately count the number of Straights beginning with Two.
Mathematically, this process of counting the variations can be written as: 4*4*4*4*4 = 1024. Any of ten cards from Ace to Ten can act as the lowest card of a Straight. By multiplying 1024 (the number of Straights with each low card) by 10, the number of possible lowest cards, we get 1024*10=10240. This figure also includes all Straight Flushes and Royal Flushes.
Total: the exact number of all possible Straights in poker is 10240 - (36+4) = 10200
Royal Flush 4
If you understand the principle of counting the number of Full Houses, you can easily understand the procedure of counting the number of five-card Three hands. Let’s start:
We remember that the number of ways to collect three different cards of one rank is
C43=4!/3!*(4-3)!=4 (first number)
Now, to make a five-card Three of kind hand, we need two arbitrary cards from the remaining deck. There are 48 of them. Again, let’s use the formula for the number of combinations of n by m.
Cnm = n!/m!*(n-m)!
We need to know how many different options of 2 cards can be collected from 48 cards.
n= 48; m=2 we have:
C482= 48!/2!*(48-2)!= 1128 (second number)
For those who do not know that in calculating C482 calculator does not work due to the fact that the value of factorial 48 has too many digits, I will say that simply by reducing the fraction this expression is reduced to 48*47/2. The calculator is useful just to find the production of 48*47 and divide by 2.
The smaller m, the simpler form we can reduce the expression, this gives this formula another advantage – the fact that you will not need a calculator in many practical cases.
To calculate the number of five-card Three of kind hands of a certain rank we should multiply (first number) by (second number):
Next, multiply by the number of ranks of cards 4512*13=58656.
We have to subtract the number of Full Houses (3744) from this number.
The number of five-card Three of kind hands is 54912.
Let’s consider Two Pairs combination.
We remember that to get two pair cards of four cards of one rank in six ways:
C42=4!/2!*(4-2)!=6 the number of options of pair cards of one rank.
The first pair may be any of six of each rank. Therefore, totally we can have 6*13 = 78 pairs of the full deck.
Now add a second pair to the first pair. The second pair can be any of the remaining twelve ranks of cards 6*12=72 The product 78*72 is the number of possible four-card combinations of Two Pairs.
The number 5616 includes recurring combinations, such as two Aces in the first pair, and the combinations having two Aces in the second pair. It is obvious that the above combinations are the same, that is, every four-card combination AA99 corresponds to a twin-combination of 99AA. To get the true number of four-card combinations of Two Pairs, we should divide the number found by two.
5616/2 = 2808
The fifth card of Two Pairs combinations can be any except the cards involved in formation of the main combination.
That is, any of forty-four 52-4-4=44
By multiplying the number of possible four-card combinations of Two Pairs by 44, we get 123552 - the number of possible five-card combinations of Two Pairs.
Now, knowing much, let’s count the number of five-card hands containing Pair.
Total, there are 6*13=78 pairs in the deck.
Let’s find how many different options of 3 cards can be collected of 48 cards:
n= 48; m=3 we have: C483= 48!/3!*(48-3)!=48*47*46/3*2*1=17296
By sorting 48 cards by 3, we did not take into account possible matches of their ranks. Therefore, from the resulting number of 1349088 we should subtract the number of Full Houses 3744, and twice the number of combinations of two pairs 123552*2=247104.
1349088-3744-247104=1098240 – the number of five-card combinations containing Pair.
Royal Flush 4
Let’s estimate the probability of collecting poker hand after dealing. As is known, the probability is the ratio of favorable outcomes to the total number of possible outcomes. The number of possible outcomes in this case is the total number of options of five-card hands that we can collect of 52 cards.
Last use and remember the universal formula for the last time:
Total five-card hand options: 2598960
Total chance to collect Royal Flush is 4/2598960=0,00000154 (In percent 0,000154)
Similarly we calculate for the rest of combinations. These data are given in Table 1.
Table 1. Probabilities of collecting the combinations after dealing in Oasis Poker
All calculations were made for the combinations collected after dealing. Probability of collecting the combinations in the box during the game depends on the rules of drawing, playing style and strategy of the player.