ROU 308: Roulette Variance

Roulette variance measures how widely a wager’s possible results are spread around its expected value. It does not measure the house edge, and it does not mean that short-term results “correct” themselves.

On a standard single-zero wheel, a straight-up bet and a red wager both carry the same 2.7027% house edge. Their session paths are very different. Red usually moves one unit at a time. A straight number loses repeatedly and occasionally pays 35 units of profit. The expected cost is the same percentage; the dispersion is not.

Expected value comes first

Let (X) be the net result of one unit wager. Returned stake is excluded from profit.

For a bet that wins (b) profit units with probability (p) and loses one unit with probability (1-p):

[ E[X]=\mu=p\times b+(1-p)\times(-1) ]

On European roulette, a straight number has:

[ p=\frac{1}{37},\quad b=35 ]

So:

[ \mu=\frac{1}{37}(35)+\frac{36}{37}(-1)=-\frac{1}{37} ]

The expected loss is 0.027027 units per unit bet, or a 2.7027% house edge.

That calculation tells us the center of the distribution. Variance tells us how far individual results tend to sit from that center.

The variance formula

For one wager:

[ Var(X)=E[(X-\mu)^2] ]

For the two-outcome roulette wager above:

[ Var(X)=p(b-\mu)^2+(1-p)(-1-\mu)^2 ]

Standard deviation is the square root of variance:

[ \sigma=\sqrt{Var(X)} ]

Variance is expressed in squared units. Standard deviation returns the measure to ordinary betting units, which makes it easier to compare with bankroll and stake size.

Standard single-zero bets compared

The following values assume a one-unit wager on a 37-pocket wheel with standard payouts.

Every row has the same expected value because standard single-zero payouts are all short by one unit across 37 possible outcomes. As coverage narrows, the win becomes less frequent and larger. Standard deviation rises sharply.

This is why “all bets have the same edge” does not mean all bets create the same bankroll experience.

A 100-spin comparison

Assume 100 independent $10 bets on a single-zero wheel.

For repeated equal wagers:

[ E[S_n]=n\mu ]

and, when the outcomes are independent:

[ \sigma_{S_n}=Stake\times\sqrt{n}\times\sigma ]

where:

• (S_n) is the total result after (n) wagers;
• (n) is the number of wagers;
• (\mu) is expected value per unit;
• (\sigma) is standard deviation per unit.

Both the red bettor and straight-number bettor have the same expected result:

[ 100\times$10\times\left(-\frac{1}{37}\right)=-$27.03 ]

But their standard deviations differ.

One $10 even-money wager per spin

[ \sigma_{100}=10\times\sqrt{100}\times0.9996\approx$99.96 ]

One $10 straight-up wager per spin

[ \sigma_{100}=10\times\sqrt{100}\times5.8378\approx$583.78 ]

The straight bettor’s standard deviation is almost six times larger even though the theoretical loss is identical. A single hit moves the session by $360 relative to a losing spin: $350 profit instead of a $10 loss.

A worse wheel does not necessarily have higher variance per unit

House edge and variance can move in different directions. On a double-zero wheel, a one-unit even-money wager wins on 18 pockets and loses on 20. Its expected value is:

[ \mu=\frac{18}{38}(+1)+\frac{20}{38}(-1)=-\frac{2}{38}=-5.2632% ]

Its standard deviation is about 0.9986 units, slightly lower than the 0.9996-unit standard deviation on a single-zero wheel. A double-zero straight bet likewise has a standard deviation of about 5.7626 units versus 5.8378 on the single-zero wheel.

That does not make the double-zero game safer or better value. The extra losing pocket shifts the average result farther toward the house while making the rare large win a little less frequent. Variance measures spread around the mean; it is not a quality score.

A favorable zero rule can change both measurements. On a single-zero even-money wager with La Partage, the possible net outcomes are +1, −1, and −0.5 units. Expected value improves to −1/74, while standard deviation is about 0.9897 units. The roulette house-edge guide should therefore be read alongside variance rather than replaced by it.

Exact hit counts matter for rare bets

A normal-distribution shortcut can be misleading over a small number of rare-event wagers. Straight-number outcomes are highly skewed.

For 100 spins betting one fixed number, the probability of no hits is:

[ P(0\ hits)=\left(\frac{36}{37}\right)^{100}\approx6.46% ]

Three hits produce a positive net result:

[ 3\times$350-97\times$10=+$80 ]

The probability of at least three hits in 100 spins is approximately 50.94%. That does not mean the bet is favorable. The losing outcomes and winning amounts still average to the same negative expectation of $27.03 for the 100-bet sequence.

A wager can have a fairly high chance of finishing ahead over one chosen session length while retaining negative expected value. Probability of profit, expected profit, and risk of a large loss are different measurements.

Stake size changes dollar variance quadratically

If a wager result is multiplied by stake (s):

[ Var(sX)=s^2Var(X) ]

and:

[ \sigma_{sX}=s\sigma_X ]

Doubling a bet from $10 to $20 doubles standard deviation in dollars and multiplies variance by four. The percentage house edge does not change.

This distinction is often missed during a progression. A player may say the “odds are the same,” which is true, while rapidly increasing the dollar dispersion and the chance of reaching a table or bankroll limit.

More spins increase total spread, but not linearly

For independent equal wagers, expected loss grows in direct proportion to the number of bets:

[ Expected\ loss=n\times Stake\times House\ edge ]

Standard deviation grows with the square root of the number of bets:

[ \sigma_{total}=Stake\times\sigma\times\sqrt{n} ]

After four times as many spins, expected loss is four times as large, while standard deviation is twice as large. Relative to total action, the random spread becomes smaller over long samples. That is one reason long-run results move toward theoretical pricing as a proportion of turnover, even though dollar swings can continue growing.

Faster roulette therefore affects both cost and experience. More decisions per hour create more expected loss and more cumulative volatility in the same clock time. See roulette expected loss per hour for the action-rate calculation.

Multiple bets on one spin are not independent

The square-root formula applies cleanly to independent spins. Several wagers resolved by the same ball result are correlated.

Suppose a player bets one unit on red and one unit on black on a single-zero wheel.

• On any red number, red wins one and black loses one: net 0.
• On any black number, black wins one and red loses one: net 0.
• On zero, both lose: net −2.

The combined position has:

[ E[X]=\frac{36}{37}(0)+\frac{1}{37}(-2)=-\frac{2}{37} ]

Its standard deviation is about 0.3243 units, not the result obtained by adding the two individual standard deviations. The wagers offset on nonzero outcomes and fail together on zero.

For a portfolio of simultaneous roulette bets:

[ Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y) ]

The covariance term measures how the two wager results move together. Ignoring it can badly misstate the risk of a layout containing overlapping or opposing positions.

Variance, volatility, and hit frequency

These terms are related but not identical.

• House edge is expected loss as a percentage of initial stake.
• Expected value is the average net result in money or units.
• Variance is expected squared deviation from the mean.
• Standard deviation is the square root of variance.
• Hit frequency is how often a defined winning event occurs.
• Volatility is a broader practical description of result size and frequency; it is not always published as one standardized number.

The NIST Engineering Statistics Handbook discussion of measures of scale defines variance as a squared-distance measure and standard deviation as its square root. Roulette uses the same statistical concepts; the payout table supplies the outcome distribution.

What variance does not prove

A large win does not prove the house edge disappeared. A long losing run does not prove the wheel is targeting a player. A result near expected loss does not prove the next session must be extreme.

Variance also does not make a previous result influence the next independent spin. Ten missed straight numbers make the observed session painful; they do not increase the next-spin probability above 1/37 on a fair single-zero wheel.

Actual wheel faults and result integrity are separate questions requiring evidence, not streak interpretation.

Bankroll use

Variance is most useful for planning exposure rather than predicting a result.

Before a session, identify:

1. total stake per spin, including every chip;
2. likely number of spins;
3. bet family and its standard deviation;
4. table minimum and maximum;
5. maximum acceptable loss;
6. whether simultaneous wagers overlap or offset.

A bankroll that feels comfortable for $10 red bets may be inadequate for $10 straight numbers, even though the expected loss per spin is the same. The roulette bankroll-risk guide and variance simulator help translate the distribution into session scenarios.

Variance explains why roulette can produce a winning night, a severe drawdown, or a result nowhere near average. It does not provide a way around the payout table. The house edge sets the long-run center; variance determines how violently individual sessions move around it.