math poker hands combinations
The calculation of the poker hands is not very difficult. There are only ten basic combinations in total that have great logic. Read this article about the poker hands and you can get started right away! We will explain for each poker hand how the hand is composed, and then why this hand is better than another.
First for the overview:
Kind of hand Percentage chance 1 in how many hands?Royal flush 0.000154% 1 in 649,740Straight flush 0.0013% 1 in 72,193 handsFour of a kind 0.024% 1 in 4,164 handsFull house 0.144% 1 in 694 handsFlush 0.197% 1 in 508 handsStraight 0.392% 1 in 255 handsThree of a kind 2.11% 1 in 47 handsTwo pair 4.64% 1 in 21 handsOne pair 42.3% 1 in 2.36 handsHigh card 50.1% 1 in 1.99 handsThe order of the poker hands is known, we start with the royal flush and end with the high card. But why is one hand better than the other? How did the game theorists calculate this?
Actually the answer is very simple. As soon as you have a small chance of making a particular hand, you have a better hand. And to calculate that probability with a deck of 52 unique cards, we can use simple probability theory .
Math behind the poker hands, the formulaThe math formula to calculate a probability is:
Total number of good opportunities------------------------------------------ * 100%Total number of possibilities
The total number of possibilities is always the same: we have to take 5 random cards from 52 different cards. To calculate the number of possibilities, we simply do a combination of 52 over 5. That's 2,598,960 different ways to draw 5 random cards.
Finally, we just need to calculate how many possibilities there are to get a particular poker hand and multiply that by 100%.
As usual, we go in order from best hand (least chance) to worst hand (best chance to get)
Royal Flush - 1 in 649,740We start with the Royal Flush. The number of possibilities to get it is very easy to calculate. 4! It is only possible with the four colors, there is no further possibility. So the formula becomes 4 / 2,598,960 * 100% = 0.000154% . That's 1 in 649,740 hands!
Straight Flush - 1 in 72.193Second, the Straight Flush. You can make nine streets for a single color. 5, 6, 7, 8, 9, 10, J, Q and K high straights. A of course not: that would be a royal. Since we have four suits, there are 36 ways to make the straight flush. That makes the formula 36 / 2,598,960 * 100% = 0.00139% . So you get a straight flush 1 in 72,193 hands.
Four of a kind - 1 in 4.165Then the four of a kind. To make the quads yourself, only four cards are needed: because they must be the same four cards, there are 13 combinations to make the quads. However, since we assume 5 cards, there are 52 - 4 (we have already used 4 cards for the quads) = 48 possibilities to get the last card.
Then we simply need to calculate 13 * 48 = 624 and we have the number of possibilities. 624 / 5,598,960 * 100% = 0.024010 . That means you get quads 1 in 4,165 times.
Full house - 1 in 694The full house gets a little more complex. First we need to calculate the number of possibilities that the first three cards are the same. First, we can choose from 13 cards. We then have to take two more from the same card. We choose three of the four colors: 4C3 = 4 combinations. So for the first three cards there are 13 * 4 = 52 different possibilities.
For the next two cards, we can first choose from 12 cards (All cards except the one we had the first three cards for). Finally, we have four suits available for those two cards: 4C2 = 6 combinations. So for the last two cards there are 12 * 6 = 72 possibilities.
Finally, we multiply the number of possibilities of the first three by the number of the last two. 52 * 72 = 3,744 possibilities to make a full house. Now we divide this by the number of possibilities and voila! 3,744 / 5,598,960 * 100% = 0.144% . That's 1 in 694 times. Those chances are getting better!
Flush - 1 in 508We calculate the flush in a slightly different way: we do not start from the number of possibilities but calculate the probability of each card immediately:
The first card doesn't matter: 52/52
The second card must be of the same suit: 12/51
The third to the fifth also: 11/50, 10/49 and 9/48
To calculate the probability of hitting a flush, we multiply all of these fractions. 52/52 * 12/51 * 11/50 * 10/49 * 9/48 = 0.00197. That's a 0.197% chance . So you get a flush 1 in 508 hands.
Straight - 1 in 255At first glance it seems very difficult to get the number of possibilities a straight: but it is actually very simple. What you need to realize is that there are ten different straights (5, 6, 7, 8, 9, 10, J, Q, K, A). So 10 Straights.
Any of the five cards can consist of any suit. We calculate that with 4 to the power of 5.
4 ^ 5 * 10 = 10240 ways to make a straight. However, this also includes all royal and straight flushes. There were 4 + 36. So the number of ways to make a normal straight is 10200.
So the formula becomes 10,200 / 5,598,960 * 100% = 0.392% . That's a 1 in 255 chance .
Three of a kind - 1 in 47The three of a kind is calculated about the same as the full house. We had already seen there that there are 52 (13 * 4) possibilities to have the first three cards the same.
For the other two cards we have to choose two different ones. (Two of the same would make a full house) Out of the 12 card taker we take 2. A combination of 12 over 2 is 66 combinations. Both cards can be in 4 suits -> 4 ^ 2 = 16 combinations of suits. So for the last two cards there are 66 * 16 = 1056 possibilities.
So we have 52 combinations for the first 3 cards and 1,056 for the last two. So there are 52 * 1.056 = 54.912 possibilities to make three of a kind. Finally only the formula: 54,912 / 5,598,960 * 100% = 2.11% . The chance that you will get three of a kind is then 1 in 47.
Two pair - 1 in 21Two pairs is a bit more complicated to calculate. First we need to see how many possibilities there are to calculate the two values of the pairs. Of the 13 values, there are 2. 13C2 = 78 possibilities.
Then we have to choose two colors from both pairs. There are of course four colors available. So we perform the combination of 4 over 2 twice. 4C2 ^ 2 = 36. The number of possibilities to get the first two pairs is therefore 78 * 36 = 2808.
For the last card, there are 44 left. (Anything but the 8 of the pairs) So the number of possibilities for the first 4 is 2808 and the last card 44. 2808 * 44 = 123.552.
123,552 / 5,598,960 * 100% = 4.75% . That makes the chance that you get two pair in a hand 1 in 21 . So you get that quite often!
One pair - 1 in 2.36As mentioned, the chance that you will get a pair is very high. First, there are 13 different cards to choose the pair from. We have four of that card and we must have two. 4C2 is 6. So for the pair there are 13 * 6 = 78 possibilities.
The next three cards must be 3 different from 12 cards. We can choose from 4 suits for each of these cards. We therefore multiply 12C3 by 4C1 ^ 3. 12C3 = 220 and 4 ^ 3 makes 64. So for the last three cards there are 220 * 64 = 14.080 possibilities.
We combine this with the first pair and find that there are a whopping 78 * 14,080 = 1,098,240 different ways. That's a chance of 1,098,240 / 5,598,960 * 100% = 42.3%. That means that you get a pair once per 2,364 hands: almost half of all hands.
High card - 1 in 1.99The high card is the worst hand of all. This is therefore very easy to get. It takes us a long time to calculate the chance that you will receive a high card. It is much easier to calculate the probability that we will not get all other poker hands . We do this by subtracting all added percentages from 100%.
The chance that you will hit something else is 49.9%. We subtract that from 100% and find 50.1%. The chance that you only get a high card is therefore more than 50%! That means that you get a high card 1 in 1.99 hands . So you have more than half of your hands with absolutely nothing.
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