Two different questions
When someone describes a game, two separate questions are usually being blended together:
- On average, how much does the game keep? This is the house edge, the long-run average loss per unit bet.
- How widely do individual results vary around that average? This is volatility, sometimes called game variance or risk.
They are independent properties. A game can have a low edge and high volatility, or a high edge and low volatility. Confusing them leads to a lot of poor reasoning, such as assuming that a game that pays often must also be cheap, or that a game with huge occasional wins must be generous.
A clean example: two roulette bets
European roulette lets us hold the edge constant while changing everything else. Compare a 1 unit bet on red with a 1 unit bet on a single number.
Red:
- Wins with probability 18/37 (48.65%), paying 1 unit.
- Loses with probability 19/37, costing 1 unit.
- Expected value: (18 - 19)/37 = -1/37, about -2.70% per unit.
Single number:
- Wins with probability 1/37 (2.70%), paying 35 units.
- Loses with probability 36/37, costing 1 unit.
- Expected value: (35 - 36)/37 = -1/37, about -2.70% per unit.
The edge is identical. The experience is not. The red bet wins nearly every other spin and rarely produces a dramatic swing. The single-number bet loses on about 97 spins out of 100 and then occasionally returns 35 units at once.
Measuring volatility: standard deviation
Volatility is usually measured with the standard deviation, which summarizes how far results tend to land from the average. The concept is explained in the article on variance and standard deviation. For one unit bet:
- Red: the results are +1 or -1, so the standard deviation is essentially 1 unit (about 0.9996).
- Single number: results are +35 or -1. The mean square outcome is (1/37 x 35 x 35) + (36/37 x 1 x 1) = (1,225 + 36)/37 = about 34.08. Taking the square root gives about 5.84 units, since the mean squared (about 0.0007) is negligible.
Standard deviation grows with the square root of the number of bets, while the expected loss grows in proportion to the number of bets. For 100 one-unit bets:
| Expected loss | Standard deviation | |
|---|---|---|
| Red | 2.70 units | about 10 units |
| Single number | 2.70 units | about 58 units |
Both lose the same 2.70 units on average. But the single-number player’s result could easily land tens of units either side of that average, while the red player’s range is much narrower.
Volatility decides how long the edge takes to show up
The expected loss accumulates steadily, but the noise around it is much larger at first. Over 100 bets, the single-number bettor’s standard deviation (about 58) is more than 20 times the expected loss (2.70). The edge is a small drift hidden inside a large swing, so short-run results look random and often show a profit.
With more bets, the drift grows faster than the noise. After 10,000 one-unit single-number bets, the expected loss is 270 units and the standard deviation is about 584, so the swing is still larger than the average but the ratio has narrowed. This is the law of large numbers in practice: the average per bet converges, even though the total can stay unpredictable for a long time.
Why volatility feels like a different game
Players often confuse the following with a “better” or “worse” game:
- Hit frequency: how often a bet pays anything. Red hits about 49% of the time, a single number 2.7%. High hit frequency usually means lower volatility, though a game can pay often in small amounts and still carry a high edge.
- Maximum win: a large top prize indicates potential for large swings but says nothing about the average.
- Bankroll swings: a high-volatility game can wipe out a small bankroll quickly even when its edge is low, which is the idea behind the risk of ruin.
The practical result is that the same budget may last a long time on a low-volatility bet and vanish quickly on a high-volatility one, even though the average cost per unit wagered is identical.
Slots, video poker and blackjack
The same two properties describe every game, though the numbers differ:
- A slot machine’s RTP sets its edge, while its volatility depends on how the return is spread between small frequent wins and rare large ones, as the slot mechanics article explains.
- Video poker’s edge depends on the pay table, while its volatility depends heavily on how much of the return comes from the royal flush.
- Blackjack under basic strategy has a small edge and moderate volatility, since results are near even-money but with doubles and splits adding variation.
Designers tune the two independently. A game could be built to keep the same margin while feeling steady or feeling dramatic. That choice is a question of variance in design, and it can be made separately from the margin.
Reading the two numbers together
When comparing games, ask for both figures. The edge answers how much, on average, will this cost per unit wagered. The volatility answers how bumpy will the road be, and how likely is a run far above or below average. Neither guarantees an outcome. A high-volatility game can leave a player ahead after a session, and a low-volatility game can still lose steadily, but over enough bets the edge decides the direction on average. Outcomes remain uncertain, and no combination of the two numbers turns a negative-expectation game into a profitable one.