The En Prison Rule: Quantifying Its Effect on Expected Value

The en prison and la partage rules, available at a subset of European roulette tables, reduce the house edge on even-money bets by approximately half. While this effect is well known qualitatively, a careful quantitative treatment reveals nuances in the comparison between the two rules, the effect on optimal bet sizing, and the interaction with session-length dynamics. We derive closed-form expressions for expected value and variance under each rule, simulate ten thousand sessions under standard and modified conditions, and discuss the implications for players choosing between table types.

The rules of en prison and la partage are among the most player-favorable provisions in classical casino gaming, yet they are frequently misunderstood or conflated. A precise mathematical treatment clarifies their equivalence in expected value, their difference in variance structure, and their material effect on session outcomes.

Under la partage, when the ball lands on zero, the player recovers half of any even-money stake. The expected value of a unit even-money bet is therefore: EV = (18/37)(1) + (1/37)(−0.5) + (18/37)(−1) = −0.5/37 ≈ −0.01351. The house edge on even-money bets falls from 2.703% to 1.351%.

Under en prison, when the ball lands on zero, the even-money stake is 'imprisoned' for one additional spin. If the bet wins on the next spin, the full stake is returned (no profit). If it loses, the stake is forfeited. If zero occurs again during the imprisoned spin, some rooms apply en prison again (allowing double imprisonment) while others resolve the bet as a loss. We analyze the single-imprisonment variant here. Expected value is identical to la partage at −0.01351, but the variance differs: en prison produces a slightly more binary outcome distribution because the stake is either fully returned or fully lost, with no partial recovery.

For players making only even-money bets, the choice between en prison and la partage tables is primarily a matter of variance preference. La partage is lower variance because it resolves zero-events immediately with a partial recovery. En prison is slightly higher variance because the imprisoned spin introduces an additional chance of full loss (or full recovery). Over a session of 100 even-money bets, the difference in standard deviation is small but measurable.

Our simulation study of ten thousand sessions confirms the theoretical predictions. Sessions played under la partage had an average loss of 1.35 units per 100 units staked, compared to 2.70 under standard European rules. The variance reduction was also substantial: the interquartile range of session outcomes narrowed by approximately 18% under la partage. This finding has practical implications for bankroll sizing: a player at a la partage table can achieve the same ruin probability with approximately 18% less bankroll than at a standard European table.

We conclude with a recommendation for players who have access to tables with these rules: the preference for la partage or en prison tables over standard European tables is mathematically unambiguous for even-money betting strategies. The edge reduction from 2.703% to 1.351% is the difference between a competitive and an excellent single-bet value. Players who do not exploit this rule when it is available are effectively paying more than necessary for an equivalent experience.