Two numbers for any repeated choice
Whenever a game offers the same random choice again and again, two numbers describe it well. Expected value is the average result per attempt over a very large number of attempts. Variance measures how widely individual results scatter around that average.
Neither number predicts the next outcome. They describe the shape of the whole pattern, which is what makes them useful for comparing options, whether the option is a bet, a skill choice or a piece of equipment.
Expected value
To find expected value, multiply each possible result by its probability and add the products.
Illustration (a made-up game): you stake 1 unit on a die roll. If it lands on 6 you receive 5 units profit; otherwise you lose your 1 unit.
- Win 5 with probability 1/6: contributes 5 × 1/6 = 5/6.
- Lose 1 with probability 5/6: contributes −1 × 5/6 = −5/6.
- Expected value = 5/6 − 5/6 = 0.
That is a fair game: on average nothing is gained or lost per play. It does not mean you break even in any one session. It means the long-run average tends toward zero.
Now a real textbook case. On a European-style roulette wheel with 37 pockets, a single-number bet typically pays 35 to 1. Staking 1 unit:
- Win 35 with probability 1/37: 35/37.
- Lose 1 with probability 36/37: −36/37.
- Expected value = 35/37 − 36/37 = −1/37, about −0.027 units per unit staked, or roughly −2.7%.
This negative figure is the house edge for that bet, and it is a feature of the payout structure. Games with a negative expected value for the player lose money on average, and no betting pattern changes the sum for independent bets. A fuller look at this idea is under negative expected value.
Expected value in game choices
The same arithmetic applies outside casinos. Suppose an attack deals 10 damage, with a 20% chance of a critical hit that doubles it to 20.
- Expected damage = 0.2 × 20 + 0.8 × 10 = 4 + 8 = 12.
So on average the attack does 12 damage per use, 20% more than the base 10. Compare it to a plain attack dealing a guaranteed 11: the first has higher expected damage, but its individual hits vary and the second never does. Which is better depends on whether you care only about the average or also about the spread, as in a fight where you need exactly enough damage to finish an enemy.
Variance and standard deviation
Variance answers “how far from the average are typical results?” The recipe:
- Subtract the expected value from each result.
- Square each difference.
- Multiply each squared difference by its probability and add them up.
The square root of the variance is the standard deviation, which is in the same units as the results and easier to picture.
Illustration with the fair die game above (expected value 0):
- Result +5 with probability 1/6: (5 − 0)² × 1/6 = 25/6.
- Result −1 with probability 5/6: (−1 − 0)² × 5/6 = 5/6.
- Variance = 25/6 + 5/6 = 30/6 = 5.
- Standard deviation = √5, about 2.24 units.
For comparison, the roulette single-number bet has a variance of about 34.1 (computed as (35² × 1 + 1² × 36)/37 − (1/37)², which is 1261/37 minus a tiny amount) and a standard deviation of about 5.8 units. Results are far more spread out than in the fair die game, even though the expected loss per bet is small.
What the two numbers say together
Take 100 single-number roulette bets of 1 unit. Expected total result: 100 × (−1/37), about −2.7 units. Standard deviation of the total: roughly √100 × 5.84, which is about 58 units, because variances add for independent bets and the standard deviation grows with the square root of the number of bets.
That is a striking comparison. The expected loss is under 3 units, but ordinary swings are around 58 units either way. Over 100 bets the luck of the sequence dominates the average, which is why a few players come out ahead and many do not, and why short-term results say little about the underlying odds. This is the heart of short run results and the way game volatility is described in design terms.
Over many more bets the average per bet settles nearer the expected value, as the law of large numbers describes, but the total loss grows in step with the number of bets. A small percentage edge, repeated enough times, wins reliably for the house.
Common misreadings
- “Expected value is what I will get.” It is an average over many trials, not a prediction for the next one, and often not even a possible result: a 12 average damage might never appear as a single hit.
- “Zero expected value means safe.” A fair game can still swing sharply, as the variance of 5 above shows.
- “Past results change the numbers.” They do not for independent events. Believing otherwise is the gambler’s fallacy.
- “Low variance is always better.” Not necessarily; it depends on the goal. Variance is a description, not a verdict.
Using them well
When comparing two repeated options, ask first for the expected value of each, then for the spread. Equal averages with different variances feel very different to play. And where money is involved, remember that expected value in a casino game is negative, uncertainty is real, and nobody can promise a result. Only stake what you can afford to lose, and contact a local support service if gambling is affecting you or someone close to you.