More Poker Math Made Simple

In my last article I introduced a simple mathematical operation called combinations that can help you to derive all sorts of probabilities at the poker table. This math is simple, especially so since you can use Google to do most of it for you, and it will help you answer fun questions about how often things happen.

As a quick review, let’s say you wanted to figure out how many possible flops there are in hold’em. There are 52 cards in the deck and three of those will be on the flop, so we need to know how many combinations of three cards you can make from 52. To find this number, you do the mathematical operation “choose” as in 52 choose three. If you want to know how to do the math of the “choose” operator, you can read my last article. But you can type “52 choose three” into Google, and it will tell you that there are 22,100 possible flops.

Pick A Board Card

There’s a popular prop bet where one player chooses one or more card ranks and wins if a card of a chosen rank appears on the flop. So a player could choose, say, a seven, and would win every time a seven appears on the flop. Any flop without a seven would be a loss.

How often does a card of any particular rank hit the flop?

The easiest way to figure this out is to calculate the negative—how often does no seven hit the flop? In general, it’s easier to calculate probabilities where the word “and” is implied rather than “or.” For example, “Will a seven hit the flop?” could be reworded as, “Will the first card be a seven or will the second card be a seven or will the third card be a seven?” Thinking about the problem as “or’s” complicates the math.

Instead you can think about it as, “Will the first card not be a seven and will the second card not be a seven and will the third card not be a seven?” This is much easier math. There are 48 non-sevens in the deck, so the total number of flops without a seven is just 48 choose three.

Typing this into Google, we get 17,296. There are 22,100 total flops, so the chance no seven appears is 17,296 / 22,100 or about 78 percent. That means there’s about a 22 percent chance that at least one seven will appear. If you could get someone to lay you 4:1 it would be great to bet on the seven. Alternatively, you could lay someone 3:1 against their card and do well.

You could change the bet to any two ranks (i.e., any seven or deuce) and calculate the probability a similar way. Now there are 44 valid cards, so the answer is 44 choose three which is 13,244, divided by 22,100. This is 59.93 percent chance that neither card appears, which makes it 40.07 percent that one of the two ranks appears. This makes 3:2 very close to fair odds.

How Likely Is A Bigger Flush?

Let’s say you’re playing no-limit against a fairly loose and reckless player. On the river you make a flush with your Q 9 on a 7 4 2 3 10 board. You get into a raising war with your opponent, and he finishes it by making a huge all-in raise. This player is loose and not-so-sharp, so your assessment is that he almost certainly has a flush, but he could potentially have any flush in the deck. What is the chance you have the better flush?

The deck starts with 13 spades, and five of them are already accounted for (your two and the three on board). That leaves eight spades total, so he can make a flush hand eight choose two ways. Typing that into Google, we get 28 possible flush combinations.
The flushes that beat you are ones that includes the ace or the king. Because this formulation uses the word or, it’s a hint that it’s probably easier to calculate the opposite. Let’s figure out how often his first card is below a queen and his second card is below a queen.

Of the eight outstanding spades, six of them are below a queen. He needs to have two of these, so there are six choose two possibilities. That’s 15. So the chance you have him beaten is 15 /28 or 53.6 percent. He’s got you beaten the other 47.4 percent of the time. If your assumptions are correct you should call his raise and hope for the best.

If you held J 9 instead, you would beat only five choose two or 10 hands out of 28, which is only 35.7 percent of combinations. In this situation whether to call or not would depend on the pot odds.

The Power Of Blockers

Blockers are a favorite poker strategy buzzword these days. The idea is that if you hold one of the key cards to make your opponents hand, it reduces the chance your opponent is holding that hand. This can be a useful guide for when to bluff, when to bet for value, and so forth—holding a blocker, be aggressive, but without a blocker, be defensive.

But how much exactly does a blocker really matter? So what if you have a club and there’s three clubs on board, your opponent can still have a flush, right?

Let’s say there are three low clubs on board, and you are hoping your opponent doesn’t have a flush. You think he would play any two suited cards preflop, so he can potentially have any combination of flush available.

There are 10 clubs outstanding, and he needs two, so there are 10 choose two or 45 possible flushes your opponent can hold.

Now let’s say you have one of the clubs in your hand. This reduces the number of possible flushes to nine choose two or 36. When you hold the blocker, your opponent’s chance of having a flush drops by nine out of 45 or by 20 percent.

It is therefore very possible for your opponent still to have a flush even when you hold a blocker, but at the same time, a 20-percent chance reduction is quite significant. Edges in this game are often thin, and having secret knowledge that your opponent is 20 percent less likely than normal to have a flush is powerful information. Players who use blocker information to inform their decision-making are often making the most of their situations.

Final Thoughts

If you play live small- and medium-stakes no-limit, then you probably don’t have to go overboard delving into the math of the game. Most of the edge at these levels comes from identifying particularly bad players and developing your strategy to take the maximum advantage of them.

But it is worth learning how to do some of the basic math of the game. It’s really not that difficult once you get used to it, and it can help make you a better player as well as just have more fun with the game. ♠

Ed’s newest book, The Course: Serious Hold ‘Em Strategy For Smart Players is available now at his website edmillerpoker.com. You can also find original articles and instructional videos by Ed at the training site redchippoker.com.