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# How does edge affect your chance of winning at roulette?

24 October 2011

The chance a roulette wager will win or lose on any round is determined solely by the number of positions on the wheel and how many of them are covered by the bet. Money “straight-up,” on one spot, at a 38-position double-zero table has one way to win and 37 to lose, so the odds to be overcome are 37-to-1. At a 37-position single-zero table, the figures are one way to fly and 36 to flop, with odds being 36-to-1. The payoff is 35-to-1 in either instance. Figures for alternate bets are analogous. A 12-number “dozen” bet represents odds of 26-to-12 or 2.17-to-1 at double-zero tables and 25-to-12 or 2.08-to-1 at their single-zero counterparts. Payoffs are 2-to-1 on either.

The odds of winning straight-up and dozen propositions are representative of the range roulette buffs typically encounter with the combinations of bets on which they risk their rubles. Whatever the wagers, edge has no impact on the likelihood of winning any individual spin. It enters the picture through the net effect of payoffs on multiple coups. This, because the edge is inherent in solid citizens being short-changed relative to the odds on successful wagers.

Pretend, for instance, that Darren and Sharon both play roulette, he double- and she single-zero versions. Further, say they each put \$1 on number 17 for 1,406 spins – roughly nine four-hour sessions – and by a quirk of fate, outcomes happen to have the statistically-correct distribution.

Darren will accordingly hit 37 times, picking up \$35 x 37 = \$1,295 and bidding adieu to \$1 x (1,406 - 37) = \$1,369. His net will be a \$1,369 - \$1,295 = \$74 deficit, not coincidentally the amount represented by the edge (5.26 percent) on the \$1,406 gross wager. Sharon will hit 38 times, gaining \$35 x 38 = \$1,330 and dropping \$1 x (1,406 - 38) = \$1,368; her net is a loss of \$1,368 - \$1,330 = \$38, also not by accident the edge (2.70 percent) times the \$1,406 gross wager.

Imagine, instead, that these intrepid individuals pin their hopes on dozen assemblages such as columns and again get the statistically-correct distributions of results. Darren would hit (12/38) x 1,406 = 444 times to snag \$888, and sacrifice \$1 x (1,406 - 444) = \$962 on the misses, for a net loss of \$962 - \$444 = \$74 – again, 5.26 percent of his \$1,406 gross wager. Sharon would take in (12/37) x 1,406 = 456 times for a gain of \$912 and give out \$1 x (1,406 - 456) = \$950, finishing down by \$950 - \$912 = \$38 – as with the straight-up bet, 2.70 percent of her \$1,406 gross wager.

A player is not apt to show a profit after 1,406 coups with these or any other roulette bets. However, 1,406 trials is a lot of roulette action; most fanciers of the felt focus on the fruits of a single session or casino visit rather than the long term. A more practical inquiry therefore might involve the chances associated with 40, 80, or 120 spins – one, two, or three hours, respectively.

The expected or theoretical loss per dollar up for grabs due to edge in 40, 80, and 120 spins would be \$2.10, \$4.20, and \$6.30 at double- and \$1.08, \$2.16, and \$3.24 at single-zero tables. These figures are the mean or average values; a player has a 50 percent shot at being deeper in the soup or of doing better – losing less or finishing even or ahead – when the dust settles, regardless of the details of the wager. The specifics of the bets, however, determine the prospects associated with the amounts by which bankrolls are up or down relative to the average.

Darren’s chance of breaking even or earning a profit betting straight-up in the double-zero game is 36 percent after 40 spins, 23 percent after 80, and 14 percent after 120. Sharon’s probabilities with the same proposition at a single-zero table are 43 percent after 40 spins, 36 percent after 80, and 29 percent after 120. Trends are similar but outlooks gloomier for wagers on dozens. Here, Darren’s chance of surfacing or soaring is only 6 percent after 40 spins, 0.1 percent after 80, and virtually nil after 120. Sharon fares a tad more optimistically, with the potential of being even or in the money equal to 22 percent after 40 spins, 6 percent after 80, and 1 percent after 120.

The impact of edge on chance of earning a profit can be seen by comparing the projections for the versions of the game with the different house advantages. The outlook, as indicated above, is much brighter at tables on which the house extracts a lower percentage. Also, for either level of edge, the promise is less unfavorable for bets with long than short odds.

On the flip side of the coin, a mathematical advantage doesn’t guarantee a rewarding session. Say, for instance, a casino offered a promotion in which it paid \$38 for a winning straight-up bet at double-zero roulette. Players would have 2.63 percent edge. Their expected earnings after 40 spins would be 1.05 per dollar at risk. The probability patrons would be at or above the neutral point after 40 spins would be 57 percent, still leaving 43 percent chance of finishing in the hole. As a reminder of this reality, recall this rhyme by the venerated versifier, Sumner A Ingmark:

A victory’s not always savored,
By gamblers when their bets are favored.