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26 October 1998

The casinos rate Boris, Doris, and Frank as \$15 bettors. They're considered equivalent as far as their action is concerned. Each of the intrepid trio does average \$15 per round. But they differ from one another in how they "spread" their bets.

* Boris bets \$15 flat, rarely raising or lowering the amount.
* Doris starts betting \$10, then jumps to \$20 if she's either ahead and thinks she's "playing with their money" or behind and hopes to catch up. Over a typical session, half her bets are at \$10 and half at \$20; the average is \$15.
* Frank starts at \$5, goes to \$10 if he smells a run, then quickly leaps to \$25 if the moon is full or other omens are auspicious. Overall, a fourth of his bets are at \$5, a third at \$10, and the rest at \$25; the average works out to \$15.

For the sake of comparison, suppose they all play the same game the same way except for their bet spreads. Make it baccarat, with bets on Player - although the reasoning applies as well to any wagers on this or other games. Are the casinos right in holding these solid citizens equivalent? Or, do the differences in betting styles somehow affect their results? Despite the apparent contradiction, the answer to both questions is "yes."

The casino accountants think the gamblers are equivalent because house earnings normalize with large samples. Many players making many bets. Some winning, others losing. Some betting big, others small. But, with enough action, money dropping into the casino bosses' coffers is the average bet multiplied by the house edge.

Say a table books \$4,500 worth of action on Player at baccarat in 300 rounds. In any combination that adds up. Boris with \$15 each time. Doris with 150 bets at \$10 and 150 at \$20. Frank with 75 bets at \$5, 100 at \$10, and 125 at \$25. Some of this and some of that. The house's theoretical win based on 1.2 percent edge is \$54. And, if the actual take isn't near \$54 after \$4,500 is bet in 300 rounds, you can be confident it'll be within a whisper of \$540,000 after \$45,000,000 is bet in 3 million rounds.

Individually, the players' results differ. The reason is their betting spreads influence the bankroll swings likely to occur in sessions of reasonable length. The fluctuations depend both on average bet and on what math mavens call "standard deviation." Players who vary their bets, for the same average, experience larger standard deviations than those who keep them steady.

For the representative spreads used by the \$15 bettors in this example, standard deviations are \$15.00 for Boris, \$15.81 for Doris, and \$17.32 for Frank. These figures can be envisioned as characteristic steps by which bankrolls change in each round.

More significantly, statisticians employ standard deviations to anticipate the range over which bankrolls can be expected to vary. Larger standard deviations mean wider ranges. For the same average bet and number of rounds, the greater the spread, the more the standard deviation and the higher or lower a player can expect to swing during a session.

Here are the ranges for the three betting styles being analyzed.
* Boris has 90 percent chance to finish between a \$480 loss and a \$360 win after 300 rounds.
* Doris has 90 percent chance to finish between a \$510 loss and a \$390 win after 300 rounds.
* Frank has 90 percent chance to finish between a \$555 loss and a \$435 win after 30 rounds.
Further, if they don't quit before 300 rounds are complete, the players also have 90 percent chance to peak twice as far ahead or behind as the indicated end points sometime during the session.

So, if you're comfortable playing at some level, choose whether and how much you want to spread your bets around the average. It's always a roller coaster ride. But spreading influences the heights and depths you're apt to pass before you get off. Sumner A Ingmark, the bankroll balladeer, said it bluntly:

Bets that, when won, make bankrolls soar,
Are those, when lost, you most deplore.

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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.