Out of the graciousness of my heart (and the promise of brownies), I decided to help some friends of mine with some probability work for their stats class. Since a few of them play cards, I decided to use poker as an analogy (and the chance to maybe win some petty cash). As we were teaching the non-poker players the various winning hands, I heard one of them ask what the chances of getting a Royal Flush was and to my astonishment one of the poker players responded “Like a million to one”.
WRONG
So I decided to give them a crash course in counting cards. Not counting cards as in the get yourself arrested for being an idiot in Vegas, but counting cards as in the probability of certain hands occurring given 52 cards. Before we can start, I’ve got a few quick notes for you. A deck has 52 cards divided into 4 suits (Hears, Diamonds, Clubs, Spades) and each suit has 13 ranks (Ace, 2-10, Jack, Queen, King). Finally, instead of the standard mathematical combination notation I’m going to use (n,k) or n choose k. If you’re not sure what a combination is, Wikipedia has a decent article.
Royal Flush: probably the easiest to figure out. There are 4 suits giving us 4 Royal Flushes total.
Straight Flush: There are 9 possible starting ranks (if we start with a 10 it’s a Royal Flush). Choose a rank (9,1) Then choose the suit (4,1) Total: 36
Four of a Kind: Choose the 2 ranks in the hand (13,2). Pick which rank occurs 4 times (2,1). Pick the suit of the remaining rank (4,1). Total: 624
Full House: Choose the 2 ranks in the hand (13,2). Pick which rank occurs 3 times (2,1). Pick the suit of the first rank (4,3) pick the suit of the second rank (4,2). Total: 3744
Flush: Pick the suit (4,1) then pick the 5 ranks (13,5) then subtract the Royal Flushes and Straight Flushes. Total: 5108
Straight: There are 10 possible starting ranks now since we can start with a 10. Choose one (10,1). Choose the suit of each rank. (4,1)(4,1)(4,1)(4,1)(4,1) Then subtract the Straight and Royal Flushes. Total: 10,200
Three of a Kind: Choose 3 ranks (13,3). Choose which is a triple (3,1) Choose suits for the triple and then the other two cards (4,3)(4,1)(4,1). Total: 54,912
Two Pair: Choose 3 ranks (13,3) Choose which two rank are paired (3,2). Choose suits for each pair (4,2)(4,2) Choose the suit of the last card (4,1). Total: 123,552)
One Pair: Choose 4 ranks (13,4) Choose which rank is paired (4,1) Choose suits for the pair (4,2) Choose suits for the leftovers (4,1)(4,1)(4,1) Total: 1,098,240
Crap: There are two ways to figure the crap hands out. One is to figure it out in a fashion similar to those above, or we could simply take the total number of hands { (52,5) = 2,598,960 } and subtract the good hands listed above for a Total of 1,302,540
So the real odds of getting a Royal Flush are 4 out of 2,598,960 or 0.00000153908% of the possible hands.
If you’d like to check my math the actual formula for n choose k (also known as a Binomial Coefficient) is n!/[k!(n-k)!] where n is the total number of objects and k is the number of objects to be chosen.
Update:
I’ve since been told that Wikipedia has an article on poker probability that goes into considerable more depth than mine. So if you’re feeling up to it head over there for more math awesomeness.
……………………..I got the Star Wars reference.