Poker is one of the most popular card games in the world, with millions of players enjoying the game both online and offline. One of the most common forms of poker is five-card draw, where each player is dealt five cards and can then discard and replace any number of them to try and make the best possible hand.
But just how many different five-card draw hands are there? Let’s take a closer look.
To start with, it’s important to understand that there are 52 cards in a standard deck (excluding jokers). Therefore, if we were simply dealing out five cards at random, without any concern for forming poker hands, there would be 52 choose 5 possible combinations. This equates to:
C(52,5) = 2,598,960
That’s over two and a half million possible combinations! However, in poker we’re not just looking at any old combination of cards – we’re looking for specific hands that have different ranks based on their relative strength.
The highest-ranking hand in poker is the royal flush – a straight flush from ten to ace. There are only four ways to make a royal flush (one for each suit), so the probability of being dealt one is:
4 / C(52,5) = 0.000154%
That’s incredibly rare! But what about other types of hand? Let’s look at them in order of strength.
Next up is the straight flush – any five cards of consecutive rank all belonging to the same suit. There are 10 possible sequences for each suit (e.g.
A-2-3-4-5 through to 10-J-Q-K-A), giving a total of 40 possible straight flushes. The probability of being dealt one is therefore:
40 / C(52,5) = 0.00139%
Still pretty rare! Next on the list is four-of-a-kind – any four cards of the same rank, plus one other card. There are 13 possible ranks to choose from (e.
four aces, four kings, four twos, etc. ), and once we’ve chosen our rank there are C(4,1) ways to choose which suit each of the four cards comes from, and C(48,1) ways to choose the remaining card. Therefore:
13 x C(4,1) x C(48,1) / C(52,5) = 0.0240%
Getting more common now! Next up is a full house – three cards of one rank and two of another.
To count the number of possible full houses we need to choose two different ranks (there are C(13,2) ways to do this), then choose which three cards will be of the first rank (C(4,3) ways to do this), and which two will be of the second rank (C(4,2) ways to do this). Therefore:
C(13,2) x C(4,3) x C(4,2) / C(52,5) = 0.1441%
Starting to get quite common now! Next is a flush – any five cards of the same suit that don’t form a straight flush.
There are C(13,5) ways to choose five different ranks for our flush (e. all hearts from 2-6), and once we’ve done that there are C(4,1)^5 ways to choose which suit each card comes from. Therefore:
C(13,5) x (C(4,1)^5 – 10 x 40) / C(52,5) = 0.197%
Note that we have subtracted out the number of straight flushes from our count here – there are 10 possible straight flushes for each suit, and we’ve already counted those. Next is a straight – any five cards of consecutive rank, regardless of suit.
There are 10 possible sequences for each card to be the highest in the straight (e. 10-J-Q-K-A), and once we’ve chosen our highest card there are C(4,1) ways to choose the suit for each card. Therefore:
10 x (C(4,1)^5 – 40) / C(52,5) = 0.392%
Again note that we have subtracted out the number of straight flushes from our count here. Next up is three-of-a-kind – any three cards of the same rank plus two other cards which don’t match each other or the three-of-a-kind.
three aces, three kings, etc. ), and once we’ve chosen our rank there are C(4,3) ways to choose which three cards will be of that rank, and C(48,2) ways to choose the remaining two cards. Therefore:
13 x C(4,3) x C(48,2) / C(52,5) = 2.11%
Starting to get quite common now! Next is two pairs – any two cards of one rank plus two more cards of another rank plus one more card that doesn’t match either pair or either singleton card. There are C(13,2) ways to choose which ranks will form our pairs (e.
two aces and two kings), then C(4,2)^2 ways to choose which specific cards will be in those pairs (e. ace of hearts and ace of spades), then C(44,1) ways to choose which remaining card we have left over after forming our pairs (any card except those in our pairs). Therefore:
C(13,2) x (C(4,2)^2) x C(44,1) / C(52,5) = 4.75%
Next is one pair – any two cards of the same rank plus three other cards which don’t match that rank or each other.
two aces, two kings, etc. ), and once we’ve chosen our rank there are C(4,2) ways to choose which specific cards will be in that pair, and C(48,3) ways to choose the remaining three cards. Therefore:
13 x C(4,2) x C(48,3) / C(52,5) = 42.30%
Finally we have high card – any hand that doesn’t fit into one of the above categories. There are C(13,5) possible different ranks for our five cards (e.
ace-king-jack-nine-seven), and once we’ve chosen those ranks there are C(4,1)^5 ways to choose which suit each card comes from. Therefore:
C(13,5) x (C(4,1)^5 – 10 x 40 – 10 x 9 x 8 / 3!) / C(52,5) = 50.11%
Again we have subtracted out straight flushes and flushes that would otherwise be counted as high card hands.
So in summary:
Royal flush: 0.000154%
Straight flush: 0.00139%
Four-of-a-kind: 0.0240%
Full house: 0.1441%
Flush: 0.197%
Straight: 0.392%
Three-of-a-kind: 2.11%
Two pairs: 4.75%
One pair: 42.30%
High card: 50.11%
As we can see, the vast majority of five-card draw hands are high card hands. However, there is still a reasonable chance of getting a pair or better, and if you’re lucky enough to see a royal flush you’ll know you’re in for a very good hand indeed!
