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# Playing it smart - What "expected value" reveals about the quality of a gamble

3 September 2007

"Expected value" is one of the clearest ways to gauge the quality of a monetary gamble. It's a cash equivalent dollars and cents that's widely used as a tool for making decisions in the world of business and investments; it applies to casino and other forms of betting as well. Expected value isn't how much money you actually anticipate having when the dust settles. Rather, it's a statistical term that accounts for both the chance and amount of various outcomes. You can also think of it as an average. Were a gamble repeated often enough, individual results would generally be higher or lower but they'd average around the expected value.

In the simplest cases, you find expected value by multiplying each possible monetary outcome by its associated probability and adding up all the products. To understand how you might use the concept in a business situation, imagine you own a widget factory. One of your competitors wants to retire and offers to sell you his company for \$10 million. Should you buy it?

You estimate that in two years, chances are 50 percent the investment will add \$15 million of worth to your operation, 10 percent it will increase your assets by the \$10 million you paid, and 40 percent it will contribute only \$1 million. The expected value of the \$10 million is 50 percent of \$15 million plus 10 percent of \$10 million plus 40 percent of \$1 million; this equals \$8.9 million. Expected value says this wouldn't be a good buy.

Or, maybe you figure on chances of 60 percent the acquisition will be worth \$25 million in three years and 40 percent it will be worth nothing. Expected value is 60 percent of \$25 million, or \$15 million. By this criterion, the gamble would be favorable.

The same principle holds in casinos. "Edge" is more often used to compare bets and project performance in this environment. But edge is expressed somewhat abstractly as a percentage; the cash equivalency of expected value is easier to envision.

Aside from "live" poker and the occasional game in which a progressive jackpot or depleted card deck crosses the critical point, the expected value of a casino bet is less than its face or nominal amount. The discount is what the house charges to offer the action, absorb the volatility, and give solid citizens the perks they've come to know, love, and demand.

As an illustration, make believe you play roulette and bet \$10 on a single spot. At a double-zero table, you'd have one chance out of 38 of winning \$350 and getting back your \$10. This is a probability of 1/38 you'll end with \$360. The expected value of the \$10 is therefore (1/38) x \$360 or \$9.47. At a single-zero table, the gamble is a 1/37 chance of ending with \$360 so the expected value of your \$10 is (1/37) x \$360 or \$9.73.

You may conclude that neither is a good gamble because they both represent losses in expected value, or that single-zero roulette is the lesser of the evils. Similar math for \$10 bets in other games shows that most impose smaller discounts than the 53 or 27 cents of roulette. A \$10 flat bet on Pass at craps, for example, has an expected value of \$9.86; \$10 at blackjack is worth \$9.95.

Make one or another of these \$10 bets 100 times during a session. The expected value of the \$1,000 gross you put at risk is \$947, \$973, \$986, and \$995 for the four games cited, respectively. You could finish up or down by various amounts in any of them, but the theoretical losses are \$53, \$27, \$14, and \$5. And, the less the theoretical loss, the more apt you are to emerge as a winner and the greater the earnings you're likely to have when you do. So, all else being equal, which would be the best game to play?

Of course, all else is not equal in a casino. Or in business. Volatility, utility, and a host of intangibles also enter the picture. This doesn't imply you can just ignore expected value. It suggests, instead, that you use it as a starting point and go on from there. As the bettors' bard, Sumner A Ingmark, wrote:

Do not ignore the fundamentals,
Or treat them as coincidentals.

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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.