Same 2.70% Edge, Nearly Six Times the Spread: Standard Deviation on Red Versus a Single Number

A £1 bet on red and a £1 bet on a single number at European roulette carry exactly the same house edge, 2.70%. The common conclusion is that they are therefore the same bet in different clothes. They are not. They share an average and almost nothing else, and the difference shows up in a number most players never calculate: the standard deviation.

The misconception, stated plainly

"If two bets have the same house edge, it makes no difference which one I play." The first half of that sentence is about expected value, and on expected value it is correct. Over a very large number of spins, both bets lose 1/37 of every pound staked. The second half is wrong, because a session is not a very large number of spins. A session is 50, 100 or 300 bets, and over that range the shape of the outcomes matters at least as much as their centre.

Expected value tells you where the results cluster in the long run. Variance tells you how far from that centre any one session is likely to land. A player who knows only the first number is reading half the label.

The maths, one spin at a time

Start with a single £1 bet, because every figure here can be checked by hand.

Red. 18 winning pockets out of 37. A win pays £1, a loss costs £1.

  • Expected value: (18/37 × £1) + (19/37 × minus £1) = minus £1/37, about minus 2.70p.
  • Variance: the average squared outcome minus the squared average. Every outcome is either plus 1 or minus 1, so the average squared outcome is exactly 1. Subtract (1/37) squared, which is 0.00073, and the variance is 0.99927.
  • Standard deviation: the square root, about £1.00.

A single number (straight up). 1 winning pocket out of 37. A win pays £35, a loss costs £1.

  • Expected value: (1/37 × £35) + (36/37 × minus £1) = minus £1/37, the same minus 2.70p.
  • Average squared outcome: (1/37 × 1,225) + (36/37 × 1) = 1,261/37, about 34.08.
  • Variance: 34.08 minus 0.00073, still about 34.08.
  • Standard deviation: about £5.84.

Same edge. Nearly six times the spread per spin.

Scaling up: what 100 spins looks like

Variance has a useful property. For independent bets, it adds. So the variance of 100 spins is 100 times the variance of one, and the standard deviation grows with the square root of the number of spins, not the number itself. The expected loss, meanwhile, grows in a straight line.

Spins at £1Expected resultStd dev, redStd dev, single number
100minus £2.70£10.00£58.38
1,000minus £27.03£31.61£184.61
10,000minus £270.27£99.96£583.78

Read the first row carefully. After 100 spins on red, the expected loss is £2.70, but a typical session lands anywhere within roughly £10 either side of that. On a single number, the typical spread is nearly £60. The edge is a rounding error next to the noise.

Now read the last row. After 10,000 spins on red, the expected loss of £270 is almost three standard deviations from break even. The edge has stopped being noise and become the whole story. That crossover, from variance dominating to expected value dominating, is the law of large numbers working exactly as described, through dilution rather than correction.

The counterintuitive part: who is more likely to be ahead?

Here is the calculation that most surprises people. What is the probability of finishing a session with more money than you started?

On red, you are ahead after n spins if you win more than half of them. On a single number, each hit returns £36 (the £35 win plus the stake), so you are ahead if the number of hits multiplied by 36 exceeds the number of spins. Both are binomial probabilities, computed exactly below.

Spins at £1Chance of being ahead, redChance of being ahead, single numberWhat you need on a single number
3743.4%26.4%2 or more hits
10035.5%50.9%3 or more hits
1,00018.8%45.1%28 or more hits
10,0000.3%32.5%278 or more hits

Two things stand out.

First, the single number column does not decline smoothly. At 37 spins one hit returns exactly £36 against £37 staked, so a lone hit still leaves you behind, and you need two. At 100 spins three hits are enough, and three is close to the expected count of 2.7, which pushes the probability just above 50%. The chance of being ahead depends on where the payout threshold falls relative to the spin count, not only on the edge.

Second, the high variance bet keeps a meaningful chance of being ahead far longer. After 10,000 spins, red leaves a player ahead in about 3 sessions in 1,000. The single number still does so about a third of the time.

That second point is the one that gets misread. It does not make the single number a better bet. Its expected loss at 10,000 spins is the identical £270.27. It reaches that average by combining a large number of sessions that end ahead with a smaller number that end much further behind. On red, almost every session ends close to minus £270. On a single number, outcomes are spread over hundreds of pounds in both directions, and the long losing tail pays for the winning sessions. In 100 spins, there is a 6.5% chance of not hitting the number once, a straight £100 loss, a result nearly ten red standard deviations below the expected one.

Intuition against the numbers

What intuition suggestsWhat the maths shows
Same house edge means same experienceSame average, but a single number has about 5.8 times the standard deviation of red per spin
The safer bet is more likely to leave me aheadOver 100 spins, red leaves you ahead 35.5% of the time and a single number 50.9%
Playing longer gives the results time to even outVariance grows more slowly than expected loss, so longer play makes the edge more visible, not less
A higher chance of finishing ahead means a better betExpected loss is identical; the extra winning sessions are balanced by larger losing ones

Why this matters beyond roulette

Every slot's volatility rating is a description of this same quantity. Two slots can publish nearly identical RTP figures and behave nothing alike in a 200 spin session, because one returns its average through frequent small wins and the other through rare large ones. The RTP is the expected value line in the tables above. The volatility rating is a non numerical stand in for the standard deviation column, and providers rarely publish the number behind it.

The practical consequence is about budgets, not odds. A higher variance game needs a larger budget to survive the same number of bets with the same probability, because the swings around the average are larger. Nothing about choosing a lower or higher variance bet changes the expected loss per pound staked.

What this means at the table

The house edge tells you what a bet costs on average; the standard deviation tells you how wildly any single session can miss that average, and two bets with the same edge can differ by a factor of six on the second number.

Responsible gambling

High variance bets are the ones most likely to produce a memorable win early in a session, and that early win is not evidence of anything except variance. Anyone offering roulette to players in Great Britain must hold a UK Gambling Commission licence, and play is for adults aged 18 and over. GAMSTOP provides free self exclusion from every licensed online operator, BeGambleAware offers confidential information and support, and the National Gambling Helpline is open around the clock on 0808 8020 133.

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