Roulette, and the only number that changes

A European wheel has 37 pockets: the numbers 1 to 36, half red and half black, plus a single green zero. An American wheel has 38, adding a second green pocket marked double zero. That one extra pocket is the whole difference between the two games, and it is worth almost twice as much to the operator.

Bets split into two families. Inside bets cover specific numbers and pay large: a single number pays 35 to 1, a split of two numbers pays 17 to 1, a corner of four pays 8 to 1. Outside bets cover large groups and pay small: red or black, odd or even, and high or low each pay 1 to 1, while a dozen or a column pays 2 to 1.

The striking property of a European wheel is that the choice of bet is irrelevant to the cost. Take a single number: it wins once in 37 and pays 35 plus the stake back, so across 37 unit stakes you get 36 units back for 37 staked. The shortfall is one unit in 37, or 2.70%.

Now take red. It wins on 18 of 37 pockets and pays 1 to 1, so across 37 unit stakes you win 18 and lose 19, a shortfall of one unit in 37. Again 2.70%. A dozen wins 12 times in 37 and pays 2 to 1, giving 24 units won against 25 lost: one unit in 37, 2.70% once more. Every bet on the layout returns the same figure, because every payout is calculated as though the zero did not exist.

That is the honest summary of roulette strategy: on a European wheel, no bet is better value than another. Spreading chips across the layout, covering more numbers or chasing patterns changes the shape of your results, never the cost. The only genuine decision is which table to sit at.

Add the double zero and the arithmetic shifts. A single number still pays 35 to 1, but now wins once in 38: you receive 36 units for 38 staked, a shortfall of two units in 38, or 5.26%. Red still pays 1 to 1 but wins on 18 of 38 pockets, losing on 20, which is the same 5.26%. As on the European wheel, the bets are uniform.

There is one exception, and it is the only bet in either game that breaks the pattern. The American five-number bet covers zero, double zero, 1, 2 and 3, and pays 6 to 1. It wins five times in 38, returning 35 units for 38 staked: a shortfall of three units in 38, or 7.89%. It exists because the payout was rounded down to a whole number, and it is the most expensive bet on either layout.

French roulette runs on a European wheel with an additional rule. Under La Partage, when the zero lands on an even-money bet, half the stake is returned instead of the whole stake being lost. The zero still loses, but it loses half as much.

The effect is exact and easy to check. On red, across 37 unit stakes, you win 18, lose 18, and lose half a unit on the zero: a shortfall of half a unit in 37, or 1.35%. The rule halves the cost of even-money bets. A related rule, En Prison, locks the stake for one further spin instead of splitting it, with a similar effect over time.

So the same game spans 1.35%, 2.70% and 5.26% depending only on the wheel and the table rules, a range of nearly four to one. Sitting at a French table rather than an American one is worth more than every betting system ever devised, which is not a compliment to roulette so much as a comment on betting systems. All three figures remain costs, and none of them is a route to profit.

Roulette attracts more systems than any other game, and the wheel is indifferent to all of them. Each spin is independent, so a colour that has appeared ten times running is exactly as likely to appear again as it was on the first spin. Tracking previous results tells you about the past and nothing about the next spin.

Progressive staking, doubling after losses in particular, does not help either. It converts a high chance of a small win into a small chance of a very large loss, because a losing run escalates stakes until it meets your bankroll or the table maximum. The average cost per unit staked stays at the figure set by the wheel.

One historical exception deserves mention precisely because it does not apply online. On physical wheels, mechanical imperfection could genuinely bias outcomes, and players have exploited it. Online roulette produces its result from a certified random number generator, and live-dealer roulette streams a real wheel that operators monitor for exactly this reason. Neither leaves a bias for a player to find.