The spread behind the average
Casino games are games of chance with a built-in house edge. No strategy, system, pattern or timing changes the odds in the player’s favor over time, and most players lose money over time. Nobody should gamble money they cannot afford to lose.
The expected value of a bet tells you the average result. It says nothing about how far a single result is likely to be from that average. Standard deviation (SD) measures that spread. It is the standard unit for describing how much results vary, and it explains why two games with the same house edge can feel very different.
Computing it for one bet
For a single bet, the steps are:
- Find the expected value (EV).
- Find the average of the squared results, E[X²].
- Variance = E[X²] − EV².
- SD = the square root of the variance.
The worked cases use single-zero roulette, which has 37 equally likely pockets, because the probabilities are textbook facts.
An even-money bet
A $1 bet on red wins $1 with probability 18/37 and loses $1 with probability 19/37.
- EV = 18/37 − 19/37 = −1/37 ≈ −$0.0270.
- Every outcome is ±1, so E[X²] = 1.
- Variance = 1 − (0.0270)² ≈ 1 − 0.00073 = 0.99927.
- SD ≈ $0.9996, which rounds to about $1.00.
The result of one bet is either +1 or −1, so an SD of about $1 is what we would expect. The SD is about 37 times larger than the average loss.
A single-number bet
A $1 bet on one number wins $35 with probability 1/37 and loses $1 with probability 36/37.
- EV = 35/37 − 36/37 = −1/37 ≈ −$0.0270. The same edge as before.
- E[X²] = (35² × 1/37) + (1² × 36/37) = (1225 + 36)/37 = 1261/37 ≈ 34.08.
- Variance ≈ 34.08 − 0.00073 ≈ 34.08.
- SD = √34.08 ≈ $5.84.
Both bets lose 1/37 of a dollar per dollar on average, but the single-number bet’s results are spread far more widely: an SD of about $5.84 versus about $1.00.
Many bets: the square-root rule
For repeated, independent bets of the same kind, the EV grows in proportion to the number of bets (n), while the SD grows in proportion to the square root of n.
- Total EV = n × (−1/37)
- Total SD ≈ √n × (single-bet SD)
Hypothetical series of even-money $1 bets:
| Number of bets | Expected result | SD (approx.) | EV ± 1 SD |
|---|---|---|---|
| 100 | −$2.70 | $10 | about −$12.70 to +$7.30 |
| 1,000 | −$27.03 | $31.6 | about −$58.6 to +$4.6 |
Check: 100 × 1/37 = 2.70; √100 = 10. And 1000 ÷ 37 = 27.03; √1000 ≈ 31.62, giving −27.03 − 31.6 = −58.6 and −27.03 + 31.6 = +4.6.
Compare this to the single-number bet over 100 bets. The EV is still −$2.70, but the SD is √100 × 5.84 ≈ $58.4. A range of EV ± 1 SD runs from about −$61 to +$56. The average cost is identical, and the possible experience is completely different.
What the numbers mean
A common rule of thumb is that for a large number of bets, roughly two-thirds of results fall within one SD of the EV, and about 95% within two SDs. This relies on a normal-curve approximation and is only rough for small samples or very skewed bets. It is a description of a pattern across many players or many sessions, not a promise about one session.
Looking at the even-money example, the interval for 100 bets includes positive results. Many ordinary players will finish a short session ahead, which is fully consistent with a negative EV. As the number of bets grows, the expected loss grows in proportion to n while the spread grows only with √n. In the 1,000-bet case, the average loss is 27.03 and the SD is 31.6; at 10,000 bets, the EV is about −$270.27 while the SD is $100, so a result above zero would be roughly 2.7 SDs from the average and quite unlikely. This is the law of large numbers in action, and it is why a casino with a vast number of bets sees results close to the edge, while an individual sees something much noisier.
Why it matters for players
- Short sessions are dominated by luck. The SD can be much larger than the expected loss, so the outcome mostly reflects variance, not the edge. A good short result says little about the game, and it does not indicate a pattern or a method.
- High-variance games feel different. Games that pay rarely and in large amounts, such as the single-number bet, have large SDs. Statistical variance in slots works the same way, and slot volatility is the same idea expressed in the language of machines.
- Streaks are expected. The spread explains why runs of wins or losses appear, and why random streaks are not evidence of anything.
- The edge still wins in the end. A large SD means results can stay above zero for a while. It does not alter the sign of the expected value.
Cautions
All figures here are textbook or invented. They assume independent bets and a fixed bet size. Real play has changes in bet size, which changes the picture, and a player who raises bets after losses is not following a fixed series. Standard deviation describes spread, not safety: a wide spread can bring a large loss as easily as a large gain. If you find gambling causing stress or trouble with money, a local support service can help.