Well, this article is seven days past deadline (Sorry Matt! I’ll make it a good one!), so I fi gured it’s a good opportunity to show how my math habit has really helped me get a feel for certain situations.

For those of you who don’t know, I’m a total math dork (at least when it comes to poker) and whenever I’m curious about something, I’ll either use poker programs like Poker Stove, or just sit down and solve random equations hoping to fi nd an informative answer.

Semi-bluffi ng is always a very mathematical situation. Virtually any poker situation can be translated into some form of math, but in some situations it’s easy and effi cient to do so, while in others it’s nearly impossible. So let’s examine some of those spots.

But fi rst, let’s start even simpler.

You have one pot-sized bet left (in other words, there is $100 in the pot and both you and your only opponent have $100 left). You have zero chance of winning the hand unless your opponent folds. If your opponent never calls an all-in, you should obviously bet 100% of the time. If he always calls, you should never bet. Assuming that the only bet you can make is all in, what is the mini-mum % of the time your opponent needs to fold to make a bet correct?

This is just about the simplest math (other than immediate pot-odds calculations) that you can have in poker, so I hope you are able to solve this. Give yourself a minute if needed.

Think of it in terms of risk vs. reward. You are risking $100 to win $100. In other words, when you bet and are called, you lose your stack of $100. When you bet and get a fold, you win the pot of $100. If you haven’t already fi gured out that the line is 50%, you can set up an equation (although in this case it’s much easier to use a combination of logic and guess and check). Fe (Fold equity) – Ce (Call equity) = 0. This is a pretty common formula that needs to be set up, so make sure you understand how to do it. Fe = (% of the time your opponent folds) * (net profi t) Ce = (% of the time your opponent calls) * (net loss) (Note: I defi ned Ce as the amount of money lost. It is probably standard to change the minus sign to a plus sign, and instead consider Ce to be the net money made or lost when called. I hope no one is confused since I tend to do small things like that a bit differently.) 100F – 100C = 0 100F = 100C F = C

So now we have two variables. Remember how your 8th grade math teacher taught you how to replace one with the other? Let’s replace the C in F=C. So how do we rewrite C in terms of F? Well, he either folds or calls 100% of the time, so C+F=1 and C=1-F. Combining the two equations, we get F=1-F; 2f=1; F=.5 That’s our answer. If he folds 50% of the time or more, we should bluff.

Now let’s complicate things slightly. Pretend instead that we have 25% equity when called. When we check, our opponent will bet and we will have to fold. It is important to understand why I bother saying that last sentence. If we have positive equity when we check, then the math changes drastically. Instead of setting the equation to zero, we would have to set our equation equal to our equity when we check, and that is pretty hard to estimate accurately.

Before I solve this, try to estimate an answer (without doing math). If he folds 60% of the time, should we bluff? What about 40% of the time? 30? 20? 10? I hope you realize that if we were bluffi ng before with a fold occurrence of 50%, then we certainly need to bluff if he’s folding that much this time, since we have extra equity when called (but the same equity when we check or get a fold).

Again, we have exactly the same equation – which is why I said it’s an important one to understand. Fe-Ce=0. 100F – Ce = 0. What’s our equity when called? Well, 75% of the time we lose our stack of 100; 25% of the time we win a profi t of 200 (our opponent’s stack plus the amount of money in the pot). Ce = (.75)(100) – (.25)(200) Ce = 75 – 50 = 25 100f – 25 = 0 100f = 25 F = 25/100 = .25 If our opponent folds more than 25 percent of the time, we can bet and show a profi t. Is it a big profi t? Let’s fi nd out – with math!! Isn’t learning fun? Let’s say our opponent folds 35% of the time. E = Fe – Ce E = (.35) (100) – .65[(.75)(100) – (.25) (200)] E = 35 – .65 (75-50) E = 35 – .65 (25) E = 18.75

So even if we have only a 25% chance of winning when called, and our opponent folds only 35% of the time, we are still showing a huge profi t of almost 1/5th of the pot when we move all in. This is why semi-bluffi ng is such a powerful play, even if we think our opponent is much more likely to call than to fold.

Of course, you can twist the numbers any way you like. Let’s say instead of one pot-sized bet left, we have a 2/3 PSB left. In other words, there is $300 in the pot, and both players have $200 stacks remaining. How often do you think our opponent needs to fold to fi nd our break-even point if we have zero equity when called? I fi nd that when I estimate these things before I solve for them, over time I get a much better feel for the situation and where the line is. You obviously won’t have time to do this math in the middle of a hand, so it really pays off to be familiar and comfortable with these situations in advance. 0 = Fe – Ce 0 = 300F – 200C F+C=1 0 = 300F – 200(1-F) 0 = 300F – 200+200F 200 = 500F .4 = F

In this situation, our opponent needs to fold only 40% of the time. What about if we have 33% equity when called? Can you give a good estimate to what kind of fold equity we need for our play to be break-even? Trick question! With a 2/3 pot sized bet left, we actually show a profi t even when we get called! Of course that makes the play profi table even if we are called 100% of the time. The obligatory math: (1/3)(500) – (2/3)(200) = 33.3 Of course that’s not only a decent profi t when called, but it’s an even bigger profi t (of 300) when we get a fold.

I have found that in semi-bluff situations, if you ask random poker players (even successful professionals) to estimate what percent fold equity they need to bluff, they will generally assume they need more than they really do. That is why I like to actually fi gure these things out. I’ve done hundreds of these calculations in my spare time, and I think it’s a big part of the reason I feel so comfortable in a wide array of mathematical situations.

Unfortunately, I know that most people who read this will never actually try to form these equations on their own, which is a shame. But if you take one thing away from this, it should be how much more often it is profi table to semi-bluff than it may seem, especially in situations where you think your opponent is slightly more likely to call than fold.

If, however, you are one of the few not afraid to apply 8th grade algebra to poker, I strongly recommend it at every available opportunity. You can set up simple equations or manipulate the one I used in a variety of ways. I guarantee that if you are able to use this skill and are willing to do so from time to time, it will greatly help you get a better feel for many different situations.

And to all of those people who like to play by feel and tend to concentrate on reading players rather than doing the math, I always say: Learn both skills. There is no reason to limit yourself to just one of the many aspects of poker. Sure, there is a lot more to poker than the math, but the math tends to follow you around in poker, even when you don’t know that it’s there.