Watching poker hands in most popular films and TV shows you'd be forgiven for thinking that every other hand has four of a kind or a straight flush but in actuality, these hands are extremely rare. But how do we figure out what is the rarest hand in poker?
What Are The Rarest Poker Hands?
The rarest possible made hand in poker is a royal flush. A royal flush is a five-card hand made up of the cards T, J, Q, K, and A, all of the same suit. Some poker players can go their whole lives without making a royal flush, such is their rarity.
Examples of royal flush hands includeand .
But why is it so rare? The simple answer is the number of combinations of royal flushes there are in a deck of cards. There are only 4 ways you can make a royal flush - having T, J, Q, K, and A of clubs, diamonds, hearts, or spades. If we compare this to the second rarest hand - a straight flush has 36 possible ways to make a straight flush that's lower than a royal flush which makes it 9 times more common than a royal flush.
If we compare it further still to how many millions of combinations of pairs you can make in a five-card hand it goes to show how rare a royal flush is!
Computing Poker Hand Probabilities
While it's obvious to anyone who's played poker before - or even just picked up a deck of cards - that there are only four combinations of royal flushes, how do we figure out the actual probability of getting one, or any other hand type for that matter?
A deck is made up of 52 cards meaning that for the first card of our five-card hand there are 52 to choose from. After that card has been picked there are 51 cards left meaning we have 51 to choose from for our second card, 50 for our 3rd card, 49 for our 4th, and 48 for our 5th. To get the total number of ways to make a five-card hand we need to multiply all these numbers together 52x51x50x49x48 - totaling a whopping 311,875,200 combinations!
However, while this is the total number of ways to draw out five cards, in poker the order of the cards doesn't matter. So to get the total number of poker hand combinations we need to remove the combinations that are the same poker hand in a different order. To do this we figure out how many combinations of the same hand there can be.
In a five-card poker hand, there are five cards that can be put in the 1st position. Once that card has been picked there are four that can be put in the 2nd position, three in the 3rd position, two in the 4th position, and only one to go in the 5th position. Just as before to find the total number of combinations we multiply these numbers together which totals 120.
Therefore we divide our original 311,875,200 hand combinations by 120 to get the total number of five-card poker hands:
311,875,200 / 120 = 2,598,960
We can now use this number to find the probability of making poker hands. So to find the probability of making a royal flush we take the total number of possible royal flush combinations (4) and divide it by the total number of poker hand combinations (2,598,960).
4 / 2,598,960 = 0.00000153907 = 0.000153907%
A tiny fraction of 1%, equivalent to roughly 1 in 649,740 - if you only play live poker you'd be lucky to see that number of hands over a lifetime!
However, these odds are for five-card combinations only. If you play Texas Hold'em there are seven possible cards you can use to make your hand (2 hole cards and 5 board cards), making your odds a bit better.
Poker Hand Probabilities in Texas Hold'em
So how does having those two extra cards available to us change the likelihood of us making hands? Well, let's look back at our equations and see how they change.
Instead of 5 card combinations, we're now calculating 7 card combinations so the original 52x51x50x49x48 becomes 52x51x50x49x48x47x46. Meaning, the total number of hand combinations rockets up from 311,875,200 to the staggering 674,274,182,400!
But, if you remember we have to account for the same hand combinations in a different order, and with 7 cards instead of 5 there are 5,040 combinations of the same hand in a different order (7x6x5x4x3x2x1).
So to find the total number of unique 7 card hands we divide 674,274,182,400 by 5040:
674,274,182,400 / 5,040 = 133,784,560
Now I know what you're thinking - "that's way more combinations than 5 cards - I thought you said there were better odds in Texas Hold'em!" and you're right, 133,784,560 is a much larger number than 2,598,960, but with 7 cards available and with poker hands being made of 5 cards there are a lot more hand combinations we can make.
Taking a look at royal flushes, instead of there being four possible combinations like there are in the 5 card variant, there are now 4,324 combinations we can make with 7 cards - having those extra two cards really helps!
So to calculate the chance of making a royal flush in Hold'em we take the 4,324 combinations of royal flushes that are possible with 7 cards and divide that by the 133,784,560 hand combinations:
4,324 / 133,784,560 = 0.00003232062 = 0.003232062%
Now, that number may not look a whole lot different to the number of 5 card combinations, but it's equivalent to 1 in 30,940 which is a lot more likely than before!
We can extrapolate this out for all hand types and we can see that the odds of making hands is much lower in the 7 card game of Texas Hold'em compared to a game like 5 Card Stud.
Now that we know exactly how unlikely the rarest hand in poker is to make we should all be a lot more thankful when we make them! Knowing these odds may not directly improve your game but all elite players have a deep understanding of the game's math - so it doesn't hurt to know it.
This article was published on March 16, 2021, and last updated on March 12, 2021.