Understanding Variance: Why Many Small Bets Beat One Big Bet
🎰Would you rather make one $100 bet, or make one hundred $1 bets on a game of chance? 🎰
If you're spending a weekend in Vegas, you don't want to spoil the fun by answering this question. But if you're in a quantitive field like risk management or machine learning, the fundamentals are super important to know!
Let's assume the game we're playing has perfect 50/50 odds of losing. Regardless of if we are making just one bet or one hundred, our expected winnings are the same
One large bet has a 50% chance of walking away with either $200 or $0:
(0.5 x $200) + (0.5 x $0) = $100
Many small bets can either net us $2 or $0 dollars, 100 times:
((0.5 x $2) + (0.5 * $0)) x 100 = $100
So what's the difference? The key concept here is variance. Variance is simply how much a set of numbers deviates from the average of those numbers, it's a measurement of how spread out your data is.
If we only make one bet in a game with a 50% chance of winning, we can either lose everything or double our money:
The chart above shows that the two possible outcomes in our one-bet game are, relatively, very far from the mean. We won't go through the math here, but the variance here is a big number.
But we can actually shrink this big number. Some additional mathematical proofs that, again, we won't cover in this article, show that for many independent, identically distributed trials, the average approaches a normal distribution and variance per bet becomes:
variance/N
where N is the number of games we play. This division shrinks the total variance from the mean!
Let's see how this works by building the same chart for our 100-game strategy with a probabilistic approach:
As you can see, we still expect to walk away with $100, but our risk exposure has substantially decreased. We have more possible scenarios where we win between $90 and $110 than we do were we lose everything.
This principle is why investors diversify their assets and data scientists use ensemble modeling approaches. It keeps expectations stable and reduces exposure to extreme outcomes.
If you would like to see a more mathematical breakdown of how this works, let me know!