Eight Deuces in 10 Hands

Well, here we go again: yet another column about probability and how often things happen – which gets me to thinking about a comment that my friend Mark Johnson occasionally makes. Paraphrasing Mark, "If you are a supernatural being, and the only way you can die is by being struck by lightning twice in 30 minutes, eventually you'll be struck by lightning twice in 30 minutes."



But that's not the point of this column. The point of this column is what somebody wrote in the online newsgroup rec.gambling.poker. In a public letter to me (in my role as poker room manager at PokerStars.com), he said that he knows PokerStars is rigged. It seems that he'd had a string of 10 hold'em hands, and within those 10 hands, eight of them had a deuce.



Now, it's not like poker players are ever inclined to exaggerate the bad hands they get. But taking the fellow at face value, I got to thinking about it: "Hmm, I wonder how likely it is that you'll get eight hands with a deuce in them within a 10-hand string."



Well, there are two ways to figure that. There's the analytical way and the empirical way. To answer the question analytically, you need to actually figure out what the probability is of a deuce showing up in a hand. For instance, the chance of your first card being a deuce is 1/13 (easy enough). The chance that you'll get at least one deuce in your hand is: 1/13 + (12/13 × 4/51) = .14932 (or 14.932 percent). This isn't a math-oriented column, so I won't go into the details here, but the probability of getting a deuce in exactly eight hands of a 10-hand sequence is .000008048 (if you're interested in the math for this calculation, e-mail me at [email protected]).



So, that's one way to do it. But math is really not my strong suit (as I had to get help from my friend James Kittock to do the math above), and I prefer to do it empirically; that is, simulate the problem on a computer – and do it 10 million or a hundred million times. Assuming you've set the problem up correctly, you almost certainly will arrive at the correct answer, or an approximation so close that you won't even notice any error. For instance, to check that I'd correctly computed the probability of getting a deuce in my hand, I wrote a trivial little computer program to generate 500 million random hold'em hands. I had the program count how many had a deuce in them. In one of the tests I ran, 74,658,782 had a deuce; that's 14.93176 percent. I ran the test again, and this time 74,642,982 of the hands had a deuce – 14.93 percent. Note that my analytical calculation said 14.932 percent. In short, my simulation results adhered to the theory very nicely. I did a simulation of the entire "eight hands with a deuce in a 10-hand string" problem, simulating 500 million 10-hand sequences. The results varied from .00000813 to .00000827.



Now, back to the original problem: What are the chances of getting a deuce in eight of 10 successive hands? Both my direct analysis and simulation indicated that the chances are about .0008 percent (eight ten-thousandths of 1 percent), which is not very likely. However, PokerStars deals tens of millions of poker hands a day (remember, every time we deal a "game" of poker, nine or 10 people get poker hands). Just to be very conservative, I ran a bunch of simulations in which I generated one million 10-hand sequences, and checked to see how many of them had eight or more hands with a deuce in it. The results ranged from five to 16 hands. And that's based on simulations of less than the number of 10-hand sequences that PokerStars will deal in a single 24-hour period.



In short, PokerStars is dealing (through sheer random shuffling) perhaps a dozen of those sequences with "eight deuces in 10 hands" every day. In fact, if we didn't deal such a sequence in that many hands, there would be ample cause to worry about the quality of our shuffling algorithm. But, my correspondent from RGP has put my mind at ease that that particular little oddity popped up at least once.



And while we're here, I set up a simulation to see how often we'd deal the nastiest of all possible hold'em bad beats – the 989-1 nightmare. For instance, you have K-K and your opponent has Q-Q. The flop comes K-2-2 and all the money goes in. Your opponent must hit two running queens (that is, the last two in the deck) to beat you. There are 45 unseen cards (52 less two in each of your hands and three on the board). Thus, there are 45 × 44 = 1,980 possible sequences of turn and river cards. However, half of those are duplicates of each other (it doesn't matter if the turn-river comes Q Q or Q Q). So, there are 990 possible unique combinations of turn and river cards. Only one of those (Q-Q) causes you to lose the hand. Here's a sobering thought: A site the size of PokerStars probably delivers a handful of those every day. Of course, the planets have to be aligned just right to make the situation that grim (it takes a special kind of poker player to get all of his money in on the flop, drawing at exactly one two-card sequence). But it probably happens enough that that magic two-card sequence comes at least once or twice a day.



This brings me back to Mark Johnson's comment. His point, of course, is that no matter how unlikely an event is, if you wait long enough, eventually that event will happen. Here in the world of online poker, we've sped up time, so even relatively unlikely events happen all the time.



Thanks for reading.

Lee Jones is the poker room manager for PokerStars.com and is the author of the best-selling book Winning Low Limit Hold'em, currently in its third edition.