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Casinos let you have the odds, but not the edge

29 August 2011

Casino aficionados commonly confuse the meanings and implications of odds and edge. They’re related, but not the same things. The element that connects and differentiates them is the payoff.

Odds describe the chance of an event. Strictly, odds are stated in a form like “3-to-1" – meaning three ways to achieve one result versus one way for the other. The same prospect is also often and more mathematically expressed as a probability – a fraction in words or arithmetic notation, a decimal, or a percentage such as “one out of four,” 1/4, 0.25, or 25 percent, respectively.

Edge denotes the statistical fraction of money wagered by players that’s earned by the house. It’s given as a percentage. Edge information is applied theoretically to each decision. This, regardless of who wins or loses and despite the fact that the amount in question doesn’t overtly change hands. Edge does, however, reliably reflect average results over large numbers of transactions.

The distinction and the relationship between odds and edge can be illustrated with examples in roulette. Picture a “straight-up” bet on one spot at single-zero table. The wheel has 37 positions, one of which will win and the other 36 will lose. The odds players fight on this proposition are therefore 36-to-1. Expressed as a probability, this is one out of 37 and 1/37 is 0.027 or 2.70 percent. Wins pay 35-to-1. Given enough play, any selected number can be expected to win an average of once every 37 tries. Say that 37,000,000 spins are tallied. This means 36,000,000 losses and 1,000,000 wins. At a dollar a shot, the house will pick up $36,000,000 and pay solid citizens $35,000,000. The casino nets $1,000,000. That’s an average of $1,000,000 divided by 37,000,000 decisions, for a quotient of $0.027 which is 2.70 percent of each bet. This is the edge. Its being numerically equal to the 2.70 percent chance of a hit is a coincidence and not relevant.

At single-zero roulette, the odds can change substantially depending on how a person bets. But edge is the same in all cases. This, because the payoffs are adjusted in concert with the odds.

Pretend that instead of dropping a dollar on one spot, the bet is placed on a row of three spots. Any of the three numbers in the row will win, and the other 34 on the wheel will lose. The odds are therefore 34-to-3 and the payoff is 11-to-1. On a 37,000,000 spin tally, the law of averages gives the expected numbers of instances as 34,000,000 $1 losses and 3,000,000 $11 wins. That’s $34,000,000 for the bosses and $33,000,000 for the bettors. Again, the casino earns $1,000,000 in 37,000,000 trials, the same $0.027 per dollar bet, or 2.70 percent edge as for the single spot.

This reasoning can extend to the point where the odds favor the player. That is, more chance to win than to lose. For this purpose, say the minimum at a table is $0.50 and this much is placed on each of two 12-number columns. There are a total of only 13 ways to lose and 24 to win. So the player is favored to win by 24-to-13. The payoff on each of the columns is 2-to-1. But, if one column wins, the other loses, so the net profit for the player on the pair of $0.50 bets is $1 on the winner minus $0.50 on the loser or $0.50 in all. Back to the 37,000,000 spins. The house expects to pick up $1 13,000,000 times and pay players $0.50 24,000,000 times. That again leaves the casino with a net of $1,000,000 in 37,000,000 spins, $0.027 per spin or 2.70 percent edge.

Having different odds but the same payoffs alters the edge. Shifting from single- to double-zero roulette changes the odds of winning since there are 38 slots on the wheel where the ball can land. However, payoffs on corresponding bets in the alternate versions are the same. A bet on one spot has 37 ways to miss and one to hit. The odds players fight are therefore 37-to-1. Expressed as a probability, this is one out of 38 and 1/38 is 2.63 percent Success is still rewarded with a 35-to-1 payoff. Say the tally on this game records 38,000,000 spins at $1 each. The house will pick up $37,000,000 and pay out $35,000,000. That’s $2,000,000 in the casino coffers. And $2,000,000 divided by 38,000,000 is $0.0526 per dollar per spin or 5.26 percent. Repeating the exercise for the other bets cited previously, the result is always $2,000,000 for the house on $1 bet 38,000,000 times, for an edge of 5.26 percent. The change in odds of winning going from single- to double-zero roulette is small – 36-to-1 as opposed to 37-to-1 for a “straight-up” bet, respectively. The edge, though, differs by close to a factor of two, 2.70 versus 5.26 percent.

A similar effect occurs across alternate implementations of games like video poker, where the odds of corresponding outcomes are approximately equal but payoffs differ. At a jacks-or-better machine, the probability of a full house is about 1.15 percent. With returns of 800-for-1 on royals, 50-for-one on straight flushes, and 20-for-1 on quads, 9-for-1 paybacks on flushes yield overall returns of 98.3 percent, whose complement is a 1.67 percent house edge. If the payback on the flush drops to 8-for-1, with everything else kept the same, the chance of a flush is still 1.15 percent. But the return becomes 97.2 percent and edge is accordingly 2.81 percent – over twice as high. Another of the odds-edge enigmas leading the poet, Sumner A Ingmark, to pout:

My casino career had a hopeful beginning,
But the more that I play, I lose while I’m winning.

Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.