Demystifying Probability - Some Poker Math

Calculating probabilities isn't often that important when you're sitting at the poker table. The ones you could do in your head, you probably already know, while the ones you really need will take a bit of paper and more time than you have.



Often, the ones you'd spend the most time thinking about are the ones that matter the least, since they're close calls.



But away from the table, when you're trying to figure out how to handle certain types of situations, or just trying to figure out how bad your bad beat was, knowing how to calculate probabilities can be more helpful. And luckily, this knowledge doesn't require tremendous aptitude for math. It just takes a bit of perseverance.



What I'd like to do in this column is demystify poker probabilities a tiny bit. Most poker probability problems are really about counting, which is a simple and intuitive activity. For things like cards, you're often interested in the number of possible arrangements of the cards that satisfy some criterion (for example, it's a flush), out of the total number of arrangements. If all the arrangements are equally likely, it's as simple as counting the possible flushes, counting the total possible sets of cards, and dividing flushes by the total. If there are 10 possible boards, and five of them make your flush, that's a 50 percent chance. This kind of simple counting problem is the most basic probability problem, and it comes up in poker often. As in everyday life, there are often lots of useful ways to count things up. While some are much easier than others, getting the counts right is what's most important.



One of the most useful probabilities in hold'em, and a good exercise, is calculating the probability of hitting your draw with multiple cards to come. Let's say you're playing hold'em and have flopped a four-flush in hearts, and you want to know your chances of hitting it. You've probably been walked through this calculation before, but I'd like to use it to illustrate a few important points about probability calculations. First, let's walk through the calculation itself – not once, but a few different ways.



The most straightforward way would be to figure out how many possibilities there are for the remaining boards, and how many of those make your flush. There are 47 cards you haven't seen, and nine of them are hearts. So, there are 47 possible cards for the turn, and for each turn card, 46 possible river cards. That makes 47 × 46, or 2,162, possible turn/river combinations. How many of those make our hand? There are three possibilities: just the turn is a heart, just the river is a heart, and both the turn and river are hearts. For the turn being a heart, that's nine possible turn cards times 38 possible river cards, or 342 possible flushes. For the river being a heart, it's just the same thing reversed, so that's 342 more flushes. And for both being hearts, that's nine possible turn hearts times eight possible river hearts, or 72 more. Add all that up, and you have 756 possible flushes out of 2,162 possible boards, for a probability of hitting your flush of 34.97 percent.



There's a shorthand method for the above, which makes the problem conceptually easier if no easier to calculate. Combinations are unordered sets of cards: The A K and the K Aare the same combination. The notation for the number of combinations of five cards drawn from a deck of 52 is C(52,5), sometimes read as "52 choose 5," and it comes out to 2,548,980. That's the number of possible five-card boards drawn from the entire deck. Without worrying about how to calculate the number of combinations (until later), we can rewrite the above problem in terms of combinations, since the order of the turn and the river doesn't matter. So, the number of turn/river combinations is C(47,2), or 1,081. For flush boards, we need to count the two-heart boards and the one-heart boards. For the two-heart boards, that's just C(9,2), or 36. For one heart, we can skip combinations, since it's just one of each. Nine hearts times 38 non-hearts, which comes out to 342. Compare these numbers to the above, and you can see that they're basically the same, just divided by two because we're not interested in order. Divide it through and you get 34.97 percent again.



A slightly different way to go about it is to add up the probability of making it on the turn and the probability of making it on the river. For the turn, that's easy, it's 9¸47. In order to make it on the river, you first have to miss on the turn, which is 38¸47. Of those 38¸47 hands, you will then hit your flush on the river 9¸46 of the time. So, that's 38 × 9 divided by 47 × 46, or 342¸2162. Add up these two possibilities and you get (9¸47) + (342¸2162), which comes out to 34.97 percent. Basically, we've simplified things by lumping together all the times you hit on the turn. Once you hit on the turn, it doesn't matter what comes on the river.



There's a third way that's pretty popular, and seems even easier. The probabilities for all possible outcomes have to add up to 100 percent. Since you either make your flush or you don't, suppose we calculate the probability of missing your flush and then just subtract it from 100 percent. In order to miss your flush, you need a miss on the turn and on the river. That's just the probability of missing on the turn times the probability of missing on the river with the remaining cards. The turn probability is 38¸47. Once you've missed on the turn, the river probability is 37¸46. The product of the two is 0.6503, or 65.03 percent. Subtract that from 100 percent and you get 35.97 percent. There's that number again.



Lastly, we can calculate the probability of missing with combinations, which is no easier to calculate but certainly easier to express. There are C(47,2) possible turn/river combinations. And C(38,2) of them are misses – all the combinations of two non-heart cards. C(38,2) is 703, and divided by 1,081 gives you 65.03 percent.



The reason for working this simple problem through in different ways isn't to make it more complicated – it's to demonstrate that even for one of the simplest probability problems in hold'em, there are different ways to get at the right answer. All of these solutions are really doing the same thing, just expressed in a different way. In a few cases, we've divided things into cases that are easier to count separately than together. One of the solutions relies on a little trick – calculating a complementary probability and subtracting it from 100 percent. Once you have that tool in your toolbox, maybe it will come in handy again later – or maybe not. There isn't always a deductive connection between the problem and the kind of solution that will make your life easiest, and finding this kind of shortcut is a matter of experience and cleverness. But if you don't happen to think of it, there are still other ways to arrive at the right answer.



One principle that I think is well-illustrated here is that it's sometimes helpful to work the same problem through in different ways. It's easy to make small mistakes, but if you add up all the possible cases and they don't sum to 100 percent, you've done something wrong. Of course, if you're solving a problem that takes many pages of tedious calculations, working through the whole problem twice may be daunting. But you can still sanity check your numbers along the way. If your problem requires dividing the possible flops for pocket aces into six cases, make sure those six cases add up to C(50,3). If you're adding up 12 different ways to make a hand with suited connectors, and it comes to 87 percent, consider adding in the ways to miss your hand, just to make sure that comes out to 13 percent.



Calculating outs may be the last probability you wanted to hear about. After all, most hold'em guides have the probabilities and odds for hitting different draws printed in a table. In order to stand a decent chance of being able to work out a probability that you're interested in, you need two things: a varied toolbox of different types of probability and combinatoric calculations, and a willingness to pursue creative solutions. In a moment, I'll point you to a great web resource for learning about probability and poker. But first, as promised, I should explain combinations.



5 factorial, usually written as "5!", is just 5 × 4 × 3 × 2 × 1 (the 1 is optional!). The number of combinations of n cards drawn from a deck with m cards is just m! divided by n. Suppose we're trying to work out the number of possible five-card hands. That would be 52! divided by 5. This is actually a notational shorthand to combine two simple factors. First, the number of possible ordered five-card hands, which is 52 × 51 × 50 × 49 × 48. An easy way to express this without knowing the actual numbers turns out to be m, which would be 52. But in practice, if you're doing this by hand, you don't really want to calculate either m! or (m-n)!. Most of the numbers cancel out, so that you really just have to multiply the first n terms in m!. If you're interested in ordered hands, you're done. If not, we still need to divide by the number of possible orders of those five cards. That just turns out to be 5!, or in the more general case, n! So, the full expression for unordered combinations is m(m-n)! As with many things in mathematics, once you understand it, you can stop thinking about the calculations and just think about it as C(52,5), right up until you actually need the numbers. Even though it's just a bit of notational shorthand, it's notation for a concept that's important in probabilities, and that will make it much easier to work these kinds of problems through without having to keep track of too much detail.



An important caveat when adding things up is to be careful when you're counting things that may not be equally likely.



The number of distinct hold'em hands, if we care about suits, is C(52,2), or 1,326. If we want to ignore suits, we can't just divide by the possible numbers of suit combinations, because that's a different number for paired and unpaired hands. It's easier to count pairs and non-pairs separately. For pairs, there are just 13 distinct hands. For non-pairs, there are C(13,2), or 78, possible pairs of ranks, and each is available both suited and unsuited, for a total of 156 possible unpaired hands. That's 169 total. If you want to know your probability of drawing a pocket pair, it would be tempting to count up the pocket pairs (13), count up the total hands (169), divide, and come up with 7.7 percent. But a given pocket pair is less likely than a given non-pair – including suits, there are six ways to make A-A, 12 ways to make A-K offsuit, and four ways to make A-K suited. So, an easier way would be to calculate out of all possible 1,326 hands, counting suits. This turns out to be easier just because all of those 1,326 hands are equally likely. There are 6 × 13, or 78, total possible pocket pairs, which is 5.88 percent of 1,326, or about a 1-in-17 chance. There are 78 possible unpaired hands, so counting suits, that's 12 × 78, or 936, offsuit possibilities, which is 70.59 percent. And for suited hands, that's 4 × 78, or 312, out of 1,326, for a probability of 23.53 percent. Add up those three percentages and you get … 100 percent. There are no guarantees, but when things add up to 100 percent, there's a good chance we've counted everything we need to exactly once.



It's worth keeping in mind that there are poker probability problems that aren't neatly expressed in terms of counting card combinations and things like that – especially problems that involve estimated probabilities of your opponent choosing one action over another, perhaps dependent on his cards. But learning to deal with these kinds of counting problems are probably the most basic.



Finally, I'd like to recommend a web site that can be extremely helpful if you want to learn a bit more about how to carry out these kinds of probability calculations. Brian Alspach, a mathematician and poker writer, maintains a web site with a collection of his columns and other writings about poker, almost all of which concern the mathematics of poker. The columns run the gamut from basic to more advanced, and also illustrate that even with a broad variety of tools at your disposal, sometimes the calculations can get a bit messy. Here's the URL: http://www.math.sfu.ca/~alspach/.



Read a few of his columns now and then, and before too long you'll be comfortable solving some very difficult problems.

Daniel Kimberg is the author of Serious Poker and he maintains a web site for serious poker players at www.seriouspoker.com.