Video Poker: Expected Value and Variance
Another reason I really like video poker is that not only is the EV known from a simple analysis of the pay table, the variance is known as well. In this article, I will discuss variance analysis using video poker as the example. Just because I use video poker, the concept can be applied to any table game or slot machine where the EV and variance are known.
The trick to making a variance analysis easy (as easy as it can be) is to use a normal distribution curve. Now, be aware this is not perfect, but it will get us close enough. If you type “normal distribution” into google, you will get tons of great information at various levels of detail.
Side note, it’s funny to me that everyone has a tool in their pocket that has access to the majority of information known to man, yet most people use that tool to argue with people online and look at pictures of cats or porn.
A simple example of a normal distribution is tracking of a coin flip. There is some chance, albeit unlikely, that you will flip a coin 100 times and they will all land tails, however it is equally as likely that the coin will land heads 100 times. Gambling does not work like this.
Example from video poker: If I play 5 coins valued at $0.25 each, I am risking $1.25. I could win $1,000 if I hit a royal (1 in 40,000-ish chance) or I could lose $1.25 if I brick (~55% of the time) or I could win anything in between. That’s not a normal distribution, but if I repeat that game over and over and the high probability of a small loss balances out with the low probability of a big win, the patters starts to approach a normal distribution even though it never really gets there. So for long run analysis I can live with some error if for the sake of getting the tools that come along with that normal curve.
Specifically, the tools I am interested in are the standard deviations. That is, if I have a positive expected value, what are the ranges of results that are likely to occur? This helps me figure out how much money to put in my pocket so I don’t run out before I complete my objective.
Before we get to how to do a variance analysis, I think we should take a quick break here and talk about the last sentence. While my objective is to make money, being an advantage gambler means you have to come to terms with losing sometimes. Plays go tits up all the time. Don’t worry about that. What you should worry about is whether you have a positive expected value based on something you can calculate and not just what some anonymous and questionably reputable dude on the internet said. I embrace a “trust but verify approach”. The verify portion is why the majority of my gambling posts exist. I’ve heard way too many people say way too many things because “someone said”. Too often that leads to an empty wallet. I’ll do a Harley Davidson post soon on the trust but verify methodology.
Anyway, back to the variance analysis.
Step 1: Find a +EV Game.
- Let’s say someone has a full pay jacks game at their bar at the $0.25 level. I’m at 99.54%
- There is a promotion running where I get an extra $2 every time I hit a 4oak. I’m at 99.92%
- I’m going to get 0.25% in free play based on my action. I’m at 100.17% (which happens to be the same payback as full pay double bonus, but a jacks game has lower variance)
- I am going to drink 4 beers an hour valued at $2 a beer. This is what I really want, free beer! That price point somehow makes it taste better.
I’m not going to calculate the free beer in the analysis… That was just for fun. We could include the beer, but my spreadsheets are not set up to account for liquor consumption. So let’s just call the booze icing on the cake, ok? The more important thing about the beer, is that I want to be in no shape to drive when I leave, so I’m going to play for 4 hours. So here are my assumptions:
- 4 hours of play
- Playing 700 hands per hour, that’s 2800 hands.
- EV = 100.17% + beer (yummy beeeer….)
- Variance of 19.7 / Standard Deviation of 4.438…
Step 2: EV Analysis
- 2800 hands * $1.25 per hand = $3,500 coin in.
- $3,500 coin in * .17% edge = $5.95 expected profit… Might not cover the tips for the beer.
- Said another way, I expect that machine to spit out $3,505.95 in wins.
Step 3: Variance Analysis
1 Standard Deviation (68% of the time your results will fall in this range)
- Minimum Limit: 3,505.95 - sqrt2800*1.25*4.384 – 3,500 = -$287.63
- Maximum Limit: 3,505.95 + sqrt2800*1.25*4.384 – 3,500 = $299.53
2 Standard Deviation Analyses (95% of the time your results will fall in this range)
- Minimum Limit: 3,505.95 – 2*sqrt2800*1.25*4.384 – 3,500 = -$581.20
- Maximum Limit: 3,505.95 +2* sqrt2800*1.25*4.384 – 3,500 = $593.10
3 Standard Deviation Analyses (99% of the time your results will fall in this range)
- Minimum Limit: 3,505.95 – 3*sqrt2800*1.25*4.384 – 3,500 = -$874.78
- Maximum Limit: 3,505.95 +3* sqrt2800*1.25*4.384 – 3,500 = $886.68
I like having enough cash on hand for 2 standard deviations but that depends on your tolerance for risk.
Another thing to keep in mind is that the free play might not be available immediately. If that is the case, you might want to consider that when figuring out how much cash to bring.
Here are a few simulations of this game to compare my calculated results. (note: I did not include the free play in the simulations or the delicious beer).
Session 1: High and Low are within 1 standard deviation.
Session 2: High and Low are within 1 standard deviation.
Session 3: Hit a royal on this one, so this result falls in the 3rd standard deviation.