Start with the bet, not the spin
A single number lands, and the payout feels either too generous or too cruel. The calculation that matters sits underneath that feeling. Expected value is the average amount a bet returns per unit wagered when the same bet is repeated a very large number of times. It does not predict the next spin, and it does not describe what your session will look like. It describes the rate at which the mathematics grinds against the player's bankroll.
To calculate expected value for any roulette bet, you need three things: the probability of winning, the profit paid on a win, and the probability of losing. The formula is the same one used across casino games: multiply each outcome's profit by its probability, then add those products together. The result is the expected profit per unit staked.
The formula in plain terms
Expected value is built from two outcomes on a simple bet. You either win the payout or lose the stake. The formula is:
Expected value = (probability of winning × profit on a win) + (probability of losing × loss on a loss)
Profit is the payout minus the original stake, because the stake is returned to you on a win. Loss is simply the stake. The probabilities come from the wheel's layout: how many pockets produce a win divided by the total number of pockets.
Suppose, as an example, a bet pays 35 to 1 on a single number. On a European wheel with 37 pockets, the probability of winning is 1 in 37 and the probability of losing is 36 in 37. The profit on a win is 35 units, and the loss on a loss is 1 unit. The expected value is:
Expected value = (1/37 × 35) + (36/37 × −1)
The first term is 35/37, which is a little under 1. The second term is −36/37. Adding them gives −1/37. That means the average result is a loss of 1/37 of a unit per unit wagered, every spin, over the long run. The same arithmetic works for any bet on the layout, because every payout is set so that the expected value lands at the same negative fraction for that wheel.
Where the house edge lives
The expected value you calculate is the house edge expressed as a per-unit loss. The house edge and the return to player, or RTP, are complements: they sum to 100% for the same bet under the same rule set. If the expected loss per unit is 1/37, then the theoretical return is 36/37 of each unit wagered over the long run. That relationship holds for every bet on the same wheel, because the payout odds are constructed around the wheel's total pocket count.
What changes the edge is the wheel itself, not the bet you choose. A European wheel has 37 pockets with a single zero. An American wheel has 38 pockets with a single zero and a double zero. The extra pocket does not change the payouts; it changes the probability of winning. A single-number bet still pays 35 to 1 on the American layout, but the probability of winning falls from 1 in 37 to 1 in 38. The expected value calculation for that wheel gives a larger negative result per unit wagered, because the same payout is now earned less often.
| Comparison point | European wheel | American wheel |
|---|---|---|
| Pockets | 37, single zero | 38, single and double zero |
| Probability of a single-number win | 1 in 37 | 1 in 38 |
| Payout on a single-number win | 35 to 1 | 35 to 1 |
| Expected value per unit wagered | −1/37 of a unit | −1/38 of a unit, but with a lower win probability, the average loss per unit is larger |
| Effect of the extra pocket | Not present | Increases the house edge on every bet |
The table shows the mechanical difference. The extra zero pocket is a losing pocket for every bet on the layout except a bet placed directly on that zero. It adds a losing outcome without adding a corresponding winning outcome, and that is what moves the expected value further from zero.
Why a short session looks nothing like the average
Expected value is a long-run average. It says nothing about the next spin, the next ten spins, or the next hundred. In a short session, variance dominates. Variance is the distribution of outcomes around the average: how far and how often results swing away from it. A single-number bet produces a large win rarely and a small loss frequently. The average of those outcomes is the expected value, but almost no individual spin lands exactly on that average.
Suppose, as an example, a player makes 100 single-number bets of 1 unit each on a European wheel. The expected value predicts a total loss of about 100/37 of a unit, which is under 3 units. But the most common result over those 100 spins is no win at all, which is a loss of 100 units. If one win lands, the session shows a loss of 65 units. If two wins land, the loss is 30 units. The expected value is the average of all possible outcomes weighted by their probability, not the result any particular player should expect to see. The gap between the average and the typical short-run result is the variance doing its work.
Decisions per hour and total exposure
Expected value is a rate. It applies per unit wagered, so the total expected loss over time is the product of the edge, the stake, and the number of bets placed. A player who places more bets per hour exposes more total stake to the same negative expected value. Roulette is a fast game when the wheel is spinning continuously. A table with a brisk dealer and few interruptions can produce a large number of spins per hour, and each spin carries the same per-unit expected loss.
The player controls three variables inside that product: the stake per spin, the number of spins per session, and the choice of wheel. Choosing a European wheel over an American wheel changes the per-unit expected loss. Choosing a lower stake changes the total at risk. Choosing to sit out spins changes the number of bets. None of those choices changes the sign of the expected value. It remains negative on every bet on the layout.
What progression systems do
A progression system changes the stake after a win or a loss. The most common is the negative progression: after a loss, the stake increases, so that a later win recovers earlier losses plus a small profit. The arithmetic looks appealing on paper, but it does not change the expected value of any individual bet. Each bet still carries the same negative expected value per unit wagered, regardless of the stake attached to it.
What a progression changes is the distribution of outcomes. It concentrates the risk into a small number of large bets. Most of the time, the progression produces a small win, because the increased stake eventually lands on a winning number. Occasionally, it produces a very large loss, because the losing streak outlasts the player's bankroll or the table's maximum stake. The expected value of the entire progression is the sum of the expected values of its individual bets, and that sum remains negative. The system redistributes outcomes rather than removing the edge.
Suppose, as an example, a player doubles a stake after each loss on an even-money bet. The first bet is 1 unit, the second 2, the third 4, and so on. Each bet carries the same negative expected value per unit. The total expected loss of the sequence is the edge multiplied by the sum of all stakes wagered across the sequence. The sequence ends either with a win that recovers the accumulated losses plus the original stake, or with a loss that exhausts the bankroll. The average of all possible sequences is still negative, and the size of the occasional loss grows with each step in the progression.
Reading the numbers on the layout
The expected value calculation does not require memorising every payout on the layout. It requires reading the payout printed next to each bet and counting the pockets that win. The payout is always stated as a multiple of the stake. The probability is always the number of winning pockets divided by the total number of pockets on that wheel. The expected value is always the same negative fraction for every bet on the same wheel, because the payouts are set in relation to the total pocket count.
What differs between bets is variance, not expected value. A single-number bet has high variance: a large payout arrives rarely. An even-money bet has lower variance: a small payout arrives more often. The expected value per unit is identical on the same wheel. A player who prefers frequent small results chooses one bet; a player who prefers rare large results chooses another. The long-run cost per unit wagered is the same.
FAQ
Does expected value tell me what I will win tonight?
No. Expected value is an average over a very large number of spins. Any single session is dominated by variance, so the actual result can be far above or far below the expected value. The calculation tells you the rate of loss built into the bet, not the outcome of the next spin.
Is the expected value the same for every roulette bet?
On the same wheel, yes. The payouts are constructed so that every bet carries the same expected value per unit wagered. The difference between bets is variance, not expected value. A bet on a single number and a bet on red have the same long-run cost per unit on the same wheel.
Can I use a betting system to make the expected value positive?
No. A progression system changes the stake after wins or losses, but each individual bet keeps its negative expected value. The system redistributes outcomes, producing many small wins and occasional large losses, while the total expected value of the sequence remains negative.
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