What to do About Variance

Poker players tend to have a reasonable understanding of variance. We understand that if we put all our money in with the best hand at any point before the river, we still might not win. We also understand that it takes extremely large sample sizes to demonstrate anything meaningful about whether you are a winning player or not. Because variance.

As such, I initially thought that Variance would be one of the topics in my course that would be quite natural to teach with a poker framework in place. Unfortunately, it's been a bit of a challenge.

This came up on my taping of the Thinking Poker Podcast last week (the episode is still under production but I think it should be up soon). Co-host Nate suggested, somewhat in jest, that I could have my students look at the variance of their quiz/exam scores to get an idea of how their performance correlates (or doesn't) to their actual understanding of the material. Fun and interesting idea for sure, but unfortunately not something that would serve the mission of my present course.

The crux of the problem lies in the fact that this is a Probability Theory course, not an Applied Data Analysis course. If we were doing data analysis, there would be lots of things we could do with poker. For example, we could take a series of 10,000 hands from my PokerTracker database, let the quantity of interest be my winnings/losings on each hand, and calculate the sample variance. We would notice that it's huge and we could talk about what that means, such as the fact that even 10,000 hands might not be enough to claim statistically significant evidence that I am a winning player (I'm actually working on making a lab for a different class where we'll do exactly that).

But, in this class, we aren't dealing with sample variance; we are just learning about theoretical variance. The one defined as:

$$ Var(X) = \sigma^2 = \sum_{\text{all } x} (x-\mu)^2 \cdot P(X=x) $$

(I know that to some of you, this means nothing; others of you I'm sure could've written that equation down from memory yourself...)

Now, in order to say anything about the theoretical variance, you need to know the theoretical distribution of the quantity that you are talking about. To be sure, we can easily talk about the theoretical variance of an outcome from a single hand, and we have indeed done this. For example, suppose you're playing heads up and you're all-in for $100, where:

• You have K♠ 9♠
• Your opponent has A♥ 6♣
• And the board is A♣ J♠ 7♠ 5♥

So we're on the turn with one card to come, holding a spade flush draw and no other possibility for winning (a King or 9 gives you a pair, but not one that is better than your opponent's pair of aces).

Thus you will win if and only if the river is any spade, giving you 9 outs and therefore a 9/44 chance of winning. Let \(X\) represent the amount of money that you will have at the end of the hand. Your opponent has also put in $100, so at the end of the hand, \(X\) is either going to be $0 or $200. At this point of the hand, your expected value is:

$$ E(X) = \$200 \cdot (9/44) + 0 \cdot (35/44) \approx \$40.91 $$ and your variance is: $$ Var(X) = (\$200-\$40.91)^2 \cdot (9/44) + (\$0 - \$40.91)^2 \cdot (35/44) \approx \$6508.264 $$

So, that looks big. But what does it really mean? And, how do I instill in my students any understanding of either what it means or how it is useful to us?

In the process of writing this post, I actually came up with a few ideas, so I'm really happy about how laying my thoughts out here has helped me in that manner. But let me end this here and see if any of you have any suggestions. I welcome any and all comments below.

On another brief note, our topic after Variance was the Markov Inequality and Chebyshev Inquality. Although it's hard for me to remember why at this point, I do remember that these were ideas that I simply could not understand as an undergraduate. So I tried to devise an activity that walked through it all in bite size pieces. I'm pretty proud of what I came up with, so here it is. I welcome any comments and critiques on that as well.

And here's a picture of my boardwork from class illustrating an example from our textbook:

Next up: The kids have their first exam this Friday. Fall Break is next week, but I'll still try to get a post up.