Understanding Uncertainty

Poker is a game of incomplete information. You’ve no doubt heard this before, and it’s certainly true. A big part of the strategic depth of poker comes from the fact that you just never know for sure what you’re up against.

What I see discussed less is exactly how this uncertainty should affect your decision-making. First, I’ll go over a common way to discuss uncertainty in poker.

In my book, How To Read Hands At No-Limit Hold’em, I talk about how to put opponents on a range of possible hands and then use that range to calculate your best decision. You start with a range of all possible hands and use your opponents’ actions along with the basic math of combinatorics to make your best guess about what you’re up against.

This is an extremely useful skill, and if you are a little foggy about what I just wrote, I highly recommend you pick up the book as it covers all that in detail.

But the thing is, there’s a whole other layer to it. Not only do you have uncertainty about what your opponent may have because the cards are dealt face down, but you also have uncertainty about what your opponent’s strategy is. When we go through the hand reading process, we tend to make assumptions about how opponents would play hands. “Well, if he had A-K, he’d probably have raised there or bet there, and if he had A-J, he would have called there or checked there.”

These are the sorts of assumptions you have to make if you want to break the math of poker down into something that’s actually approachable.

But there’s an error in this process—an inescapable error. You do not know for certain how your opponent would play any of these hands. In fact, with most opponents you have considerable uncertainty about how they would play any hand. Even the most predictable opponents will surprise you sometimes, and the unpredictable ones—well I don’t have to tell you how inscrutable they can be.

There’s a healthy, un-removable hunk of uncertainty built into all this logic. And the thing is, this sort of uncertainty does some unintuitive things to probabilities.

To understand how this works, let’s look at a simpler problem than poker. Let’s say we’re betting on how tall the next man to walk through a door will be. For the sake of argument, say you know that the average (median) man in your city is 5 foot 10 inches tall, and that 20 percent of all men are between 5 foot 9 inches and 5 foot 11 inches. I will bet you $2 against your $3 that the next man to walk through the door is under 5 foot 11 inches.

The way we’ve defined this problem so far, that would be a fair bet. Half the men are shorter than 5 foot 10 and another approximately 10 percent are between 5 foot 10 and 5 foot 11, so about 60 percent of men are shorter than 5 foot 11. If I make you lay 3:2 odds, it’s a fair bet.

But now let’s say we go to a city somewhere else in the country. You know the heights will work roughly the same way in this new place, but you aren’t as sure about what the average is. You think it’s roughly equally likely in this new place that the average is anywhere from 5 foot 8 inches through 6 foot tall.

Is it still a fair bet to ask you to lay 3:2 that the next man through the door is shorter than 5 foot 11?

You might think so. Your estimate of the average height hasn’t changed—5 foot 10 is still your average estimate of the average. So if the average hasn’t changed, and you’re just a little less certain, the bet hasn’t changed either, right?

Obviously I wouldn’t be asking if the answer weren’t no. The bet isn’t fair anymore. Here’s why. Let’s say for simplicity that there’s a 20 percent chance of the average being any of the 5 inches from 5 foot 8 through 6 foot.

Let’s start by assuming the average still is 5 foot 10. Then the bet is still fair. But that’s the case only 20 percent of the time.

Now let’s say the real average is actually 5 foot 11. There’s now a 50 percent chance of going under, and so obviously it’s no good to lay 3:2 on it.

It’s even worse if the average is 6 foot. Now there’s about a 40 percent chance (if we assume that the 20 percent of people within an inch of average rule still holds) that the next man through the door will be under 5 foot 11.

Okay, well that part is obvious. If we’re wrong and the average person in the new place is taller, then clearly betting under is going to be worse. But what if the average is actually lower?

Well let’s say the average is 5 foot 9. We can know that 60 percent of people will be shorter than 5 foot 10. But how many will be shorter than 5 foot 11?

You can’t know for certain given the information I’ve given, but the key point is that it must be less than 70 percent. As you go further away from average, you get smaller percentages of the population. (Think of how many men are exactly 5 foot 5 or 6 foot 3. Many fewer than those who are exactly 5 foot 9 or 5 foot 11.)

So if going 1 inch away from average includes 10 percent of the population, the next inch has to be less than an additional 10 percent. For the sake of argument let’s just say it’s 7 percent. So there’s a 67 percent chance that the next man will be shorter than 5 foot 11 when the average is 5 foot 9.

Finally if the average is actually 5 foot 8, you get another inch that includes fewer people yet. Let’s say only 5 more percent. So that means there’s a 72 percent chance the next man will be shorter than 5 foot 11 if the average is 5 foot 8.

And now we can put all this together to see if the bet remains fair or not. There’s a 20 percent chance of each of the five possible averages. The chance the bet wins for each of the possibilities is 72, 67, 60, 50, and 40. Take the average of those and you get 57.8 percent. Since this is less than 60 percent, the bet isn’t fair anymore laying 3:2.

Final Thoughts

What does this have to do with poker? Because it’s a game of incomplete information, you very frequently deal with this sort of uncertainty whether you know it or not. And just like in this simple example, the amount of uncertainty you have affects the odds of winning the bets you make. Next issue I will take this idea further. ♠

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.