OKBET Poker Guide What is the Rarest Hand in Poker

A royal flush is the most difficult hand to make in poker. A royal flush is a five-card hand made up of the same-suit cards T, J, Q, K, and A.

You’d be forgiven for thinking that every other hand in most popular films and TV shows has four of a kind or a straight flush, but these hands are extremely rare. But how do we know what the rarest poker hand is?

What Are the Most Valuable Poker Hands?

A royal flush is the most difficult hand to make in poker. A royal flush is a five-card hand made up of the same-suit cards T, J, Q, K, and A. Because royal flushes are so rare, some poker players can go their entire lives without ever having one.

TJQKA and TJQKA are two examples of royal flush hands.

But why is it so uncommon? The simple answer is the number of royal flush combinations in a deck of cards. There are only four ways to get a royal flush: T, J, Q, K, and A in clubs, diamonds, hearts, or spades. When compared to the second rarest hand, a straight flush, which has 36 possible ways to make a straight flush, it is 9 times more common than a royal flush.

When we compare it to the millions of possible pair combinations in a five-card hand, we can see how rare a royal flush is!

Calculating the Probabilities of Poker Hands

While it’s obvious to anyone who’s ever played poker – or even just picked up a deck of cards – that there are only four royal flush combinations, how do we calculate the actual probability of getting one, or any other hand type for that matter?

Because a deck contains 52 cards, we have 52 options for the first card of our five-card hand. After that card is chosen, we have 51 cards to choose from for our second card, 50 for our third card, 49 for our fourth, and 48 for our fifth. To get the total number of ways to make a five-card hand, multiply all of these numbers together: 52x51x50x49x48 = 311,875,200 combinations!

While this is the total number of ways to draw five cards, the order of the cards is irrelevant in poker. To obtain the total number of poker hand combinations, we must first eliminate the combinations that are the same poker hand in a different order. We do this by calculating the number of possible combinations of the same hand.

A five-card poker hand has five cards that can be placed in the first position. After that card is chosen, four cards can be placed in the second position, three in the third position, two in the fourth position, and only one in the fifth position. As before, we multiply these numbers together to get the total number of combinations, which is 120.

To calculate the total number of five-card poker hands, we divide our original 311,875,200 hand combinations by 120:

311,875,200 / 120 = 2,598,960

We can now use this number to calculate the likelihood of making poker hands. To calculate the probability of completing a royal flush, multiply the total number of possible royal flush combinations (4) by the total number of poker hand combinations (2,598,960).

4 / 2,598,960 = 0.00000153907 = 0.000153907%

A fraction of 1%, or roughly 1 in 649,740 – if you only play live poker, you’d be lucky to see that many hands in a lifetime!

These odds, however, only apply to five-card combinations. When playing Texas Hold’em, you have seven possible cards to use to make your hand (2 hole cards and 5 board cards), which improves your odds slightly.

Texas Hold’em Poker Hand Probabilities

So, how does having those two extra cards affect the likelihood of us making hands? So, let’s go back and see how our equations have changed.

We’re now calculating 7 card combinations instead of 5, so the original 52x51x50x49x48 becomes 52x51x50x49x48x47x46. That is, the total number of hand combinations increases from 311,875,200 to a mind-boggling 674,274,182,400!

However, keep in mind that we must account for the same hand combinations in a different order, and with 7 cards instead of 5,040 combinations of the same hand in a different order (7x6x5x4x3x2x1).

So, to calculate the total number of distinct 7-card hands, divide 674,274,182,400 by 5040:

674,274,182,400 / 5,040 = 133,784,560

I know what you’re thinking: “That’s a lot more combinations than 5 cards – I thought you said Texas Hold’em had better odds!” And, yes, 133,784,560 is a much larger number than 2,598,960, but with 7 cards available and poker hands consisting of 5 cards, there are many more hand combinations we can make.

When it comes to royal flushes, instead of four possible combinations as in the 5 card variant, there are now 4,324 possible combinations with 7 cards – having those extra two cards really helps!

So, to calculate the chance of making a royal flush in Hold’em, we divide the 4,324 royal flush combinations possible with 7 cards by the 133,784,560 hand combinations:

4,324 / 133,784,560 = 0.00003232062 = 0.003232062%

That number may not appear to be much different from the number of 5 card combinations, but it is equivalent to 1 in 30,940, which is significantly more likely than before!

We can extrapolate this to all hand types and see that the odds of making hands in the 7 card game of Texas Hold’em are much lower than in a game like 5 Card Stud.

We should all be a lot more grateful when we get the rarest hand in poker now that we know how unlikely it is to happen! Knowing these odds may not directly improve your game, but all elite players have a thorough understanding of the game’s math – so it never hurts to be aware of it.

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