Expected Value of Call Bets

Why they look different

A straight upbetStraight upA single number, pays 35:1.Read full → has one outcome that pays. Voisins du zéroFR · sectorVoisins du zéro“Neighbors of zero”: the seventeen numbers from 22 to 25 on the wheel, bet with nine chips.Read full → has three different payoutsmoneyPayoutWhat a winning bet returns, quoted as “X to 1” on top of the returned stake.Read full → depending on whether the ballequipmentBallThe small ball launched along the track against the rotor. Modern balls are ivorine or Teflon, 18–21 mm.Read full → lands in the double-weighted streetbetStreetThree numbers in a row of the layout, pays 11:1. French: transversale.Read full →, the double-weighted cornerbetCornerFour numbers sharing a corner on the layout, pays 8:1.Read full → or a single splitbetSplitTwo adjacent numbers, pays 17:1.Read full →. The variety hides a constant: every chipmoneyChipA token representing a fixed amount, placed on the layout to make a bet. Also: check.Read full → in the bet is an ordinary bet with EV −1/37 of itself, and the whole is the sum.

Voisins du zéro (9 chips, 17 numbers)

2 on 0-2-3 (street, pays 11), 2 on 25-26-28-29 (corner, pays 8), 1 each on splits 4-7, 12-15, 18-21, 19-22, 32-35 (pay 17). Hit in the street: +22 − 7 = +15. Hit in the corner: +16 − 7 = +9. Hit on a split: +17 − 8 = +9. Miss: −9. EV = (3 × 15 + 4 × 9 + 10 × 9 − 20 × 9)/37 = (45 + 36 + 90 − 180)/37 = −9/37. Nine chips, −9/37: −1/37 per chip.

Tiers du cylindre (6 chips, 12 numbers)

Six splits, each pays 17. Any hit: +17 − 5 = +12. Miss: −6. EV = (12 × 12 − 25 × 6)/37 = (144 − 150)/37 = −6/37. The flattest call bettableCall betA bet announced to the dealer rather than placed by hand, usually a wheel-sector bet such as voisins or tiers. Also “announced bet”.Read full →: one payout, one loss.

Orphelins (5 chips, 8 numbers)

1 straight up on 1 (pays 35), splits 6-9, 14-17, 17-20, 31-34 (pay 17). Hit on 1: +35 − 4 = +31. Hit on 17 (two splits): +34 − 3 = +31. Hit on 6, 9, 14, 20, 31, 34: +17 − 4 = +13. Miss: −5. EV = (31 + 31 + 6 × 13 − 29 × 5)/37 = (62 + 78 − 145)/37 = −5/37.

Neighbors and jeu zéro

NeighborsbetNeighborsA number and the two pockets on either side of it on the wheel, five straight-up chips.Read full →: five straight ups; any hit +35 − 4 = +31; EV = (5 × 31 − 32 × 5)/37 = (155 − 160)/37 = −5/37. Jeu zéroFR · sectorJeu zéro“Zero game”: the seven numbers nearest zero (12, 35, 3, 26, 0, 32, 15), bet with four chips.Read full →: splits 0-3, 12-15, 32-35 and a straight up on 26; hit on 26: +35 − 3 = +32; split hit: +17 − 3 = +14; EV = (32 + 6 × 14 − 30 × 4)/37 = (32 + 84 − 120)/37 = −4/37.

On an electronic wheel

IGT’s Roulette Evolution, as Bollman (2021) describes it, offers these call bets on an electronic single-zerowheelZeroThe single green pocket that no even-money, dozen or column bet covers; the source of the house edge. Also: single zero, 0.Read full → wheel. The arithmetic does not change: each is still −1/37 per chip. What changes is the one argument players make for sectorwheelSectorA contiguous arc of pockets on the wheel, as distinct from a group on the layout.Read full → bets. At a real table some believe they can read the dealer’s habits and the speed of the spin well enough to guess the region where the ball will drop, which is called wheel clocking, and a call bet is the natural way to bet a region. An electronic wheel has no dealer to read, and that argument, weak at a real table, is gone.

Electronic Roulette Terminals

The lesson

Call bets are convenience, not mathematics. They are the fastest way to bet a wheel sector and the only way to do it in one announcement. Their expected valuemathExpected valueThe average result per bet over the long run. −1/37 of the stake for every standard European bet.Read full → is the same per chip as the plainest straight up, and the variety of payouts is bookkeeping, not edge.

Formula

Questions