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# Use Even-Money Equivalents to Compare Bets

10 May 1999

An old Bezonian proverb says you can't compare apteryxes and orangutans. Do analogous axioms apply in casinos when players try to compare games or evaluate alternate bets at the same tables?

For instance, assume you're used to playing \$1 video poker, five coins at once. You try \$5 blackjack, knowing the house edge is less than in video poker. You hit a hot table -- at least, the solid citizens with whom you're sitting tell you it is. You're risking the same amount and should have a better chance than at video poker. But, relative to the wins you've experienced when the slots were sizzling, you aren't making much money. Are video poker and blackjack like apteryxes and orangutans?

Or, maybe you relish roulette and bet \$10 a spin -- all on red or the middle 12-number column, or split as \$5 on each. You usually earn a modest profit and rarely deplete your \$200 stake. You have a friend who also plays roulette for \$10 a spin, but either bets it all on three-number rows or single spots, or splits it as \$5 on one of each. Your friend has about as many successful sessions as you -- but wins more and needs a \$1,000 buy-in, not to mention nerves of titanium, to withstand the downswings during the action. You're playing the same game and risking equal amounts, yet the dynamics you encounter differ markedly. Are bets with short and long odds like apteryxes and orangutans?

It's possible to compare different games and bets from the perspective of impact on sessions of reasonable length. This is done using "even-money equivalents." That is, the amount of and the chance you'll win a bet paying 1-to-1, which has the same statistical properties as the wager in question. I won't bore you with the math, but to satisfy any budding Nikolai Ivanovich Lobachevski in the crowd, equivalents have the same expectation and variance, with higher moments of the distribution neglected.

The following list shows theoretical even-money equivalents per dollar wagered on a variety of casino bets.

 bet equivalent amount chance of winning (%) craps place six/eight 1.08 49.30 place five/nine 1.18 48.30 place four/10 1.32 47.48 any craps proposition 2.52 47.79 three/11 proposition 3.67 48.48 two/12 proposition 5.10 48.64 double-zero roulette outside (lose half on 0 or 00) 0.98 48.66 12-number column 1.40 48.11 four-number corner 2.76 49.05 three-number row 3.24 49.19 single number 5.76 49.54 blackjack played with 0.5% house edge 1.12 49.78 played with 1.0% house edge 1.12 49.56 played with 1.5% house edge 1.12 49.33 slot machines typical 97.3% video poker 4.40 49.69 typical 95% three-reeler 7.03 49.64

Here's how to use the table to compare blackjack and video poker. Your \$5 total video poker bet on a typical machine is equivalent to \$5 x 4.40 or \$22 with a 49.69 percent chance at an even-money payout. Blackjack at \$5 with 1 percent house edge is equivalent to \$5 x 1.12 or \$5.60 with a 49.56 percent chance to win even money. Odds of winning are nearly identical in the theoretically equivalent cases, but the same nominal bet at video poker has the effect of four times as much at blackjack.

As another example, use the table to compare different roulette bets. Putting \$10 on a 12-number column is equivalent to betting \$10 x 1.40 or \$14 with 48.11 percent chance at winning even money. On a single number, \$10 is equivalent to \$10 x 5.76 or \$57.60 with 49.54 percent chance to pick up even money. A bet on one number is therefore like four times as much on a 12-number column, with the chance of winning raised by 1.43 percent.

Bets with low-probability high payoffs tend to swing more sharply than those having shorter odds. You can win more, or deplete a bankroll faster. Reduction to an even-money standard bet shows why and how much. And, it affords means to contrast different situations in anticipating the size of a bankroll needed to play as well as in estimating the size of a win it's reasonable to expect. As the poet nonpareil, Sumner A Ingmark, once proclaimed:

Comparing two equivalents,
Eliminates ambivalence.
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Best of Alan Krigman
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.