The paradoxical probabilites of poker hands
Fifty-two cards, four suits, five cards at a time…. there will be math.
A standard deck of playing cards — assuming that you’ve pulled the jokers — has 52 cards. A standard poker hand — assuming nothing wild and that you’ve got to keep the cards you were dealt — contains five cards. So, how many possible combinations of five cards are there in a deck of 52 cards?
Poker players (well, poker players who are also mathematicians) write the number of such possible combinations as 52C5. The formula is a lot more intimidating. The answer is 2,598,960. That’s how many distinct sets of five cards can be made from a standard deck of 52 playing cards.
As any poker player will tell you, some of those 2,598,960 sets of five cards are more valuable than others. Possible distinct sets, or “hands” in the lingo, are ranked based on the chances that they will be dealt. Poker hands that will come up more often in random dealing are less valuable or powerful than rarer hands.
With over 2.5 million possibilties, is it any wonder it’s called gambling?
Let’s keep assuming that we’re playing a rather dull version of poker, one where each of us is dealt five cards that we have to keep and none are wild. Here, is a list of the poker hands in the order that they’re likely to occur (from least likely to most likely) and the frequencies of them occurring. For fun, we’ve added in the formulas for calculating them too. (Feel free to double-check!):
- Straight Flush – 40 – V-3C1 × SC1
- Four of a kind – 624 – VC1 × SC4 × VS-SC1
- Full House – 3,744 – VC1 × SC3 × V-1C1 × SC2
- Flush – 5,108 – (VC5 × SC1) – (V-3C1 × SC1)
- Straight – 10,200 – (V-3C1 × SC15) – (V-3C1 × SC1)
- Three of a kind – 54,912 – VC1 × SC3 × V-1C2 × SC12
- Two pair – 123,552 – VC2 × SC22 × V-2C1 × SC1
- One pair – 1,098,240 – VC1 × SC2 × V-1C3 × SC13
- High card – 1,302,540 – (VC5 – (V-3)) × (SC15 – S)
Since the Straight Flush is least likely to occur, it’s the most powerful poker hand generally. But even that hand includes relatively mundane straight flushes like the seven, eight, nine, ten and Jack of clubs. There are 40 combinations of five consecutive cards in any suit.
The Royal Flush
Somewhat arbitrarily, there is a mostest powerful hand amongst the most powerful of poker hands: The Royal Flush. The Royal Flush is made up of the Ace, King, Queen, Jack and 10 of the same suit.
What are the odds of getting dealt a Royal Flush in our vanilla poker game where we have to keep all five cards we get and none are wild? To calculate that, we just have to know: the total number of poker hands possible (2,598,960); and the number of ways a Royal Flush can be dealt (4). Divide the second number by the first number, and that’ll tell us the probability of a Royal Flush. The answer is 4/2,598,960 or 1/649,740 or 0.00015%.
Let’s try that in context. If you dealt yourself a hand of poker every minute (twenty-four hours a day and non-stop) and the Royal Flush came last, you’d deal yourself that Royal Flush on the 452nd day!
Royal Flush, ‘Royal Shmush’
There’s a certain arbitrariness about all this. Consider this, we call the Royal Flush the most powerful hand because it’s the least likely combination to occur. Byt we call it “Royal” because it’s made up of the face cards, and the Ace. That’s all that’s special about it. But we could have made a different choice. Let’s put aside the allure of the “flush.” What if we were romanced by the allure of odd numbers? The odds of a 3-5-7-9-Jack combination in the same suit? The very same as of the odds of Royal Flush.
Or odder still, imagine an inside-out, upside-down-world where we swooned at the thought of a 2 of Clubs, 4 of Diamonds, 7 of Hearts, Queen of Diamonds, and, for a little Wild West flair, the Ace of Spades. Well, there’s only one possible version of that hand – 1/2,598,960. Imagine again, one hand per minute with that particular hand lost. You’d deal that hand on the 1,805th day — nearly five years after you began.
A deeper dive – related reading from the 101:
Poker’s a hoot, but it’s not the only game in town.
There are lots of things to be had from playing games — fun, bragging rights, competition — but empathy?