Average, Expected Value, Variance and More

We go through a few basic statistical concepts that a poker player needs to understand.

Photo courtesy of Zsuzsanna Kilian

Average

Averages are a pretty straightforward thing, and I think most of our esteemed readers have a good feel for what it means.

Roll a die a number of times, add up all results and divide by the number of rolls. Viola, you have the average.

Another word for this is mean value, it’s the same thing.

Basically, the average is the value “in the middle”. The point around which something is spread out. You could say that the outcomes of die rolls are spread out around the mean value 3.5.

In reality you cannot know beforehand which average you’ll actually get from a series of die rolls. You know that the mean value is expected to be 3.5, but in real series the mean value will be spread out around this value.

Example: When writing this article I took a die and rolled it ten times with the following result: 6, 3, 3, 1, 2, 1, 2, 6, 6, 4.

To get the average value, sum up the values (34) and divide by the number of rolls (10). You’ll find the mean value of this series to be 3.4. That’s pretty close to the expected 3.5.

Example: I did it again and got this result: 1, 6, 5, 6, 3, 2, 6, 4, 5, 4. This time the mean value, or average, is 4.2.

As you see, the mean values spread out around the expected value. They vary. That’s normal.

Expected value

The expected value of an event is called expectation value, or EV. It’s the value you should expect it to have in the long run based on the incomplete knowledge you have. Your best guess if you like.

Example: When rolling a die several times, the expected average is 3.5. You cannot be sure of it, but it’s your best guess.

In poker, you use EV all the time to make decisions about calling or folding. If the EV of a call is below zero, calling is a mistake since in the long run you expect to lose money by calling. Read more in our article on The Math behind Calling and Folding.

Variation around the average

The mean value can vary a lot between short series. If you throw a die just once, the mean value is the value of the die. It can be anything from 1 to 6, and it can’t possibly be 3.5.

However, as you perform more die rolls, the average will draw closer to the expected 3.5. For very long series of rolls, the mean value will likely get very close to the expected value, the EV.

This phenomenon is known as the law of large numbers. It says that if you repeat an experiment many times, the frequency of an event will approach the frequency that is expected from theory.

For example, if you roll a die one million times, with a very high likelihood the mean value will be close to the expected 3.5.

In this case, if the mean value turns out to be far away from the expected value, this would be an indication that there’s something fishy with the die or the way it’s being rolled.

The long run

The law of large numbers is what we’re talking about when we say that something evens out in the long run.

You can be card dead for a while, meaning that most of your starting hands come from the lower end of the spectrum. Your average is lower than the expected average. This happens to everyone all the time.

However, over time, for example after one hundred thousand hands or so, your average starting hand will draw closer and closer to the expected Q7.

Or more speicfically, the probability goes down that the average will lie outside a narrow interval around the expected value.

Long and longer runs

Since getting a starting hand is the most common event in poker, the starting hand average is probably what evens out the fastest.

Events that happen less frequently will need more time to even out. It just takes longer to collect a long series of these events.

For example, the event of shoving with aces and getting a call by kings is quite rare (once in five thousand starting hands or so). It will take much, much longer for the average outcome of this event to approach the expected average of 80% wins for the aces.

If you do this match-up one hundred times (corresponding to some 500,000 played hands), there’s still 10% probability that your winning frequency is below 75%.

Confidence intervals

Around the expected value are values that are quite likely to occur, just a little less likely than the expected value.

If you include some of these values in an interval, you have a confidence interval. Meaning: there’s a certain probability that the outcome will lie in that interval.

For example, if you roll a die ten times and look at the interval between 3 and 4, there’s a 64% probability that the average will lie in this interval. In this case, this interval is a 64% confidence interval for the average.

If you roll a die 20 times, there’s an 81% probability that the mean value will lie between 3 and 4. Now this interval is an 81% confidence interval.

We said above that when the series get longer, the mean value will approach the expected value. Even with just ten more rolls, the spread gets narrower.

If you roll a die 100 times, the mean value will be between 3 and 4 with 99.7% probability.

The confidence for the interval increases. Or, the interval containing a certain percentage of confidence gets narrower, which is the same.

Of course, in the long run we’ll all be dead. So you’d better enjoy the game while you can. Living in the present is always plus EV, they say.