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Poker Math
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Derivations for Five Card Stud
Many have inquired about how I calculated the probabilities related to various poker hands, prompting me to create this section for a clearer understanding of the mathematical processes involved. This explanation assumes a moderate understanding of mathematical concepts; if someone is comfortable with concepts typically taught in high school, they should be able to follow along without issues. The techniques discussed here can be useful for various probability-related scenarios.
The Factorial Function
If you're already familiar with the concept of a factorial, feel free to move on. However, if you're unsure and think that 5! refers to shouting the number five, then read on.
When it comes to your living room sofa, the assembly guide will likely suggest rearranging its cushions from time to time. Let's say your sofa has four cushions. How many different ways can you arrange them? The solution is 4!, which equals 24. You have 4 choices for the first cushion, followed by 3 for the second, 2 for the third, and just 1 for the last, resulting in 4 multiplied by 3 multiplied by 2 multiplied by 1, which equals 24. If you had a total of n cushions, you would calculate it as n times (n-1) times (n-2) all the way down to 1, which is represented as n!. Most scientific calculators will feature a factorial button, typically shown as x!, and in Excel, the function fact(x) can provide the factorial of a given x. If we're discussing a deck of 52 cards, the number of ways they can be arranged is 52!, which equals approximately 8.065818 × 10.67.
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The Combinatorial Function
Consider a scenario where you're forming a committee of 4 individuals chosen from a group of 10 within your workplace. What are the different combinations available? The calculation goes like this: 10! divided by (4! times (10-4)!), which results in 210 different possibilities. In general terms, if you're forming a group of y individuals from a total of x, the formula to determine the combinations is x! divided by (y! times (x-y)!). Why this method? For instance, if we have 10 people, 10! equals 3,628,800 possible arrangements. You may think of the first four as your selected committee and the others as those left out. However, since the order of selection doesn’t matter, you must account for the arrangements of the chosen committee and the remaining individuals. The 4! equals 24 arrangements of the committee members, and 6! equals 720 arrangements for those not chosen. By dividing 10! by the product of these two factorials, you only retain the actual combinations of individuals, mathematically represented as (1*2*3*4*5*6*7*8*9*10) divided by ((1*2*3*4)*(1*2*3*4*5*6)), resulting in a total of 210 distinct combinations. Excel's (x,y) function can also calculate the number of ways to select a group of y from x. combin Now, let's calculate how many unique five-card hands can be made from a standard 52-card deck. The calculation yields combin(52,5), which simplifies to 52! divided by (5!*47!), giving us 2,598,960 distinct combinations. If you're manually calculating this—perhaps your calculator lacks a factorial feature, or you don't have access to Excel—keep in mind that all factors of 47! will cancel out those in 52!, leaving you with (52*51*50*49*48) divided by (1*2*3*4*5). The chance of obtaining any specific hand can be determined by taking the number of ways the hand can occur and dividing it by the overall combinations, which is 2,598,960. Below you'll find the combination counts for each type of hand; simply divide by 2,598,960 to find the respective probabilities.
The following section outlines how to calculate the number of combinations for every type of five-card stud poker hand.
Poker Math
For a royal flush, there are four distinct ways to achieve it, one for each suit.
Royal Flush
The maximum card in a straight flush can be any one of the following: 5, 6, 7, 8, 9, 10, Jack, Queen, or King. Therefore, there are 9 potential highest cards and 4 different suits, leading to a total of 9 * 4 = 36 unique straight flush combinations.
Straight Flush
When it comes to four of a kind, there are exactly 13 possible ranks you can select from. The fifth card can be any of the remaining 48 cards, resulting in 13 * 48 = 624 potential combinations.
Four of a Kind
For a full house, there are 13 ranks from which to choose for the three of a kind, and 12 ranks remaining for the two of a kind. The three cards of one rank can be arranged in 4 ways (choosing from the four specific cards available), and combin(4,2) yields 6 ways to choose two cards of another rank. Therefore, the total combinations for a full house is 13 * 12 * 4 * 6 = 3,744.
Full House
You have 4 suits to choose from and, using combin(13,5) = 1,287 possibilities to select five cards all in the same suit. From this, subtract 10 for the high cards that could lead into a straight, which would then be considered a straight flush, leaving you with 1,277. When you multiply this by 4 for the four different suits, you arrive at 5,108 different combinations to create a flush.
Flush
For straights, the highest card can be any of the following: 5, 6, 7, 8, 9, 10, Jack, Queen, King, or Ace. This results in 10 possible high cards, and each card can be from any of the 4 suits. The number of arrangements for five cards across these four suits is calculated as 4 raised to the power of 5, equaling 1024. Subtracting 4 from 1024 accounts for the four arrangements possible that would create a flush, yielding 1020. Consequently, the total ways to form a straight becomes 10 multiplied by 1020, equaling 10,200.
Straight
For three of a kind, you can select from any of the 13 ranks, and there are 4 methods to arrange 3 cards from a selection of four. To choose the additional two ranks for the other cards, there are combin(12,2) = 66 ways. Considering each of these two ranks has four possible cards, the total arrangements for having a three of a kind is 13 * 4 * 66 * 4.5To form two pairs, there are 78 ways to choose the two ranks involved. For each chosen rank, there are 6 arrangements possible from the four cards. Meanwhile, the fifth card can be drawn from the remaining 44 cards, leading to a final tally of 78 * 6.
Three of a Kind
There are 13 ranks to select for forming a pair, coupled with combin(4,2) = 6 ways to arrange the two cards making up the pair. The arrangement of the other three ranks, each representing a singleton, can happen in combin(12,3) = 220 ways, with four cards to select from for each rank. Thus, the total combinations appear as 13 * 6 * 220 * 4.2= 54,912.
Two Pair
Initially, calculate the possibilities of selecting five distinct ranks from a total of 13—this is found by combin(13,5) = 1287. You will then need to subtract 10 for the ten different high cards that could initiate a straight, resulting in 1277 remaining options. Each card could be represented by one of 4 suits, giving 4 raised to the power of 5, or 1024 ways to arrange the suits for these 1277 combinations. Yet we need to deduct 4 from the 1024 to eliminate the four scenarios where a flush could form, ultimately leading to 1020 ways to arrange them. Therefore, the final combinations for a high-card hand concludes as 1277 * 1020 = 1,302,540.2* 44 = 123,552 ways to arrange a two pair.
One Pair
For instance, if we want to establish the probability of drawing a jack-high hand, all cards must be lower than a jack, meaning you’ll need four cards from the nine available ranks. The ways to combine those four from nine is given by combin(9,4) = 126. We then take away 1 for the specific combination of 10-9-8-7 that would create a straight. Thus, we are left with a total of 125. Since we found earlier that there are 1020 arrangements for suits, multiplying these gives us 127,500 as the total number of ways to achieve a jack-high hand. Remember, for an ace-high hand, you have to subtract 2 rather than 1 because both A-K-Q-J-10 and 5-4-3-2-A lead into valid straights.3= 1,098,240 ways to arrange a pair.
Nothing
For those interested in more on probability...5I will also be calculating probabilities related to drawing an ace-high hand where the second card is a king. The remaining three cards must be distinct and fall within the ranks of queen down to two. The number of ways to select three out of those 11 ranks amounts to combin(11,3) = 165. After subtracting one for the Q-J-10 sequence that results in a straight, we end up with 164 combinations. Following the previous method, there are again 1020 ways to arrange the suits to avoid forming a flush. Thus, the ultimate number of arrangements for the ace/king hand totals 164 * 1020 = 167,280.
Specific High Card
For numerous other poker-related probabilities, kindly refer to my dedicated section on this topic.Here is a good site that also explains how to calculate poker probabilities .
Five Card Draw — High Card Hands
| Hand | Combinations | Probability |
|---|---|---|
| Ace high | 502,860 | 0.19341583 |
| King high | 335,580 | 0.12912088 |
| Queen high | 213,180 | 0.08202512 |
| Jack high | 127,500 | 0.04905808 |
| 10 high | 70,380 | 0.02708006 |
| 9 high | 34,680 | 0.01334380 |
| 8 high | 14,280 | 0.00549451 |
| 7 high | 4,080 | 0.00156986 |
| Total | 1,302,540 | 0.501177394 |
Ace/King High
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