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# Even-Money Equivalents Help Anticipate Swings in New Situations

25 November 2003

Everybody understands that the higher the payoff on a bet, the less likely it is to hit. You may not know the precise chance of a \$50,000 slot machine jackpot with \$1 bet. But you can be sure it's smaller than the prospect of a \$5,000 win for the same wager on a similar game two seats away. Likewise, you needn't be a whiz at craps to realize that since \$30 pays \$42 on the nine and \$35 on the eight, the former must be tougher to win than the latter.

So, how do you balance a small chance at big bucks against a better shot at a more modest return? Some folks won't be happy unless they make major moolah and are bent on going for broke, even if the axe falls quickly. Others want lots of action on their stakes and target lengthy sessions, comps, and any profit at all. It's between these extremes that the wicket gets sticky.

Experience in gambling a certain way leads to intuition about how funds flow under normal conditions. How far and fast fortunes rise and fall. Changing games or betting styles alters these factors, especially when the variations involve lower or higher payoffs per dollar at risk. A solid citizen might jump in and test something new, but experiments can be expensive. Even-money equivalents provide clues for evaluating alternatives.

Even-money equivalents are sizes and probabilities of theoretical wagers paying 1-to-1, having the same impact on bankroll swings as the prototypes they represent. To see the idea, pretend you're accustomed to hazarding \$10 to win \$20 on a 12-number column at double-zero roulette. The even-money equivalent is a 48.1 percent probability of winning \$13.90 with \$13.90 on the layout.

The accompanying list gives even-money equivalents for a range of double-zero roulette options. Edges are all 5.26 percent. To show the effects of going longer, hypothetical 100-to-1 and 2,500-to-1 payouts with 5.26 percent edge are appended to the data. Reducing house advantage raises both size and probability of even-money equivalents slightly. Conversely for increasing the house's cut.

 actual payoff even-money equivalent per dollar bet probability of winning even-money equivalent 0.5-to-1 \$0.72 46.4% 1-to-1 \$1.00 47.4% 2-to-1 \$1.39 48.1% 5-to-1 \$2.19 48.8% 8-to-1 \$2.76 49.0% 11-to-1 \$3.24 49.2% 17-to-1 \$4.02 49.4% 35-to-1 \$5.76 49.5% 100-to-1 \$9.73 49.7% 2,500-to-1 \$48.67 49.9%

You can see the trends. Say you're thinking of switching from \$10 on one column (nets \$20, 2-to-1), where you have a good "feel" for the game, to splitting the \$10 into \$5 on each of two columns (nets \$5, 0.5-to-1). The new even-money equivalent is \$7.20 with 46.4 percent chance to win. Oppositely, consider \$5 on each of two spots (nets \$170, 17-to-1). The even-money equivalent is \$40.20 with 49.4 percent chance to win. The \$10 on a 2,500-to-1 shot would be like betting \$486.70 with chances of 49.9 percent.

As you opt for more extreme longshots, the probability of the even-money equivalent gets closer to 50 percent. Of course, house edge prevents the chance from ever quite reaching this limit.

The size of the even-money equivalent is trickier to estimate. But you can do it by punching one key on almost any calculator, without a PhD in arithmetic. Amount approaches the "square root" of the payoff odds. A square root is a value, which multiplied by itself, gives the original number. For instance, the square root of 25 is 5 because 5 x 5 = 25. So, enter any arbitrary payoff per dollar into your calculator and hit the square root key (it looks like a checkmark). The answer will appear on the display. If you're not sure which is the right key, find it by entering the number 144 then hitting what you guess might be the square root. Do this until the 144 changes to 12, then you've got it.

Even-money equivalents are means to extrapolate from the bankroll swings characteristic of bets with which you're familiar to those you can expect with new wagers. They don't tell what to do, but help decide what to try. As the poet, Sumner A Ingmark, put it:

Get guidance in new situations,
From models and approximations.

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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.