Probability and variance

EV tells you how much you win on average. Variance tells you how far the results stray from that average. A value bettor who ignores variance sooner or later ends up abandoning a good strategy… or going broke.

The variance of a bet

For a stake of 1 unit, the result is either $+(\text{odds} - 1)$ with probability $p$, or $-1$ with probability $1 - p$. The variance is:

The standard deviation (the square root of the variance) is more intuitive: it's the order of magnitude of the "typical" deviation from the average.

What this changes in practice

• At short odds, results quickly track the EV. Deviations are moderate.
• At long odds, you can stay in the red for a very long time with a genuine edge, then win it all back in a few hits.
• The variance of a series of $n$ independent bets adds up: the total standard deviation grows like $\sqrt{n}$, while the expected profit grows like $n$. That's why volume eventually wins out over luck.