What is Variance?

2/3 — Three Key Factors That make a Casino Strategy “Impossible” to beat.

This is the second article of the series:

  1. Expected Value (EV)
  2. Variance
  3. Bankroll management (Risk of Ruin)

In the first part of this series, we’ve discussed Expected Value and used 3 different examples of EV calculations to demonstrate what is a scientific approach to calculating the average outcome of an action in a few simple cases. Calculating the EV of a bet/investment will show us what is the expected outcome of this bet on average, which is a pretty decisive factor to take into consideration about if we want to take this bet or not.

In this second part, we are going to discuss another important factor that dictates the sustainability of a betting strategy and allows us to make scientific projections of our strategy’s future results…

After explaining the 3 examples I think that pretty much everyone can understand what makes the house/casino always be on the profitable side. It’s because the casino only allows players to place negative EV bets. This does not mean that a player who made a profit by gambling in a casino will never exist but it does mean that the casino will only be taking bets in its favor making each bet a financially wrong decision for the player every time.

Is this enough for the casino to be sustainable? How can someone be profitable in one of those games? How does “luck” affect these results?

The answer is given by the term variance. Let’s explain this term so we get a grasp of it.

The statistical definition of Variance is the average of the squared differences from the mean:

σ² = ( Σ (x-μ)² ) / N

In simple terms, variance is a measure that counts how much a result can differ from the expected outcome or how dispersed the data is about the mean. The bigger the variance, the more repetitions are required for the results to be representative (tending to EV).

The variance of a die roll (6-side fair die)

Let’s present the 6-side die example and calculate variance so we understand in practice what it is.

So what is the meaning of a die roll? Assuming the die is fair we know that each outcome is equally possible (⅙) so the mean (most people understand this term as average) would be (1+2+3+4+5+6) / 6 = 3.5.

This means that the EV of a single die roll would be 3.5. In other words, if someone asked us to roll the die X times and every time we’d roll the die we would be awarded in $ by the value of our role, we could assume that at the end of this dream scenario we would win X*$3.5 since on average the number we would roll would be 3.5 (the mean).

That being said we know that in reality, we would never get the number 3.5 in a roll, that’d be impossible since the only possible results are 1–2–3–4–5–6 and this is why most people have trouble understanding this concept. Meaning the paradox that even though the average outcome of our roll is 3.5 we’d never have this exact result appearing in our sample.

Let’s use the definition above to calculate the variance of a die roll. As we said variance is the average of the squared differences from the mean so we’d first need to find all the differences from the mean, then square those differences and sum them and finally divide them by 6 to find the average.

If we sum up the numbers in the third column we would get the number 17.5, now dividing this number by 6 we get a variance of a 1 die roll which equals = 2.91666 (variance is also often represented as σ² with σ being the standard deviation which we can see that in this case, it’s 1.707 meaning that each die roll will deviate from the mean an average of 1.707)

In plain words variance (and standard deviation) gives us a measurement of how much results differ from the mean on average thus explaining to us how big of a sample we’d need for the results to be representative.

The lowest variance is the smaller sample we’d need to be able to “judge” the results and vice versa.

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