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# Is Playing the Percentages Right for You?

19 May 1997

Pretend, as a promotional ploy, a casino let you pick one of two great gambles. Bet \$10 and you'd get a 63 percent chance to win \$10. Bet \$20 and you'd get a 60 percent chance to win \$20.

Both bets are robust because you have an edge - even-money payoffs with 1.7-to-1 odds favoring success. Of course, neither may suit you. For instance, maybe you disapprove of gambling, need the cash for gas and tolls on the way home, or got warned off by 1-900-psychic. Otherwise, your selection is between the two options. Do you prefer 63 to 60 percent success rate? Would you be swayed by the possible \$20 rather than \$10 payoff? Are you willing to lose \$10 on this proposition, but not \$20?

A fantasy? No. Solid citizens face choices like this whenever they're dealt a possible double down in blackjack. And, there's a standard answer... although it may not be right for you.

Consider five-five versus a dealer's seven-up. This is a strong hand. In a six-deck game, hitting offers a 63 percent chance to win one betting unit with a single unit at risk. Doubling affords 60 percent success rate - a 3 percent drop in the chance to win and two betting units at risk - but twice the potential profit.

Basic strategy, which indicates decisions yielding the highest expected gains over extended play, dictates doubling. And, most blackjack buffs act accordingly. Statistical analysis shows how much the extra money outweighs the lower odds of winning a particular hand. For six-deck games, expected gain for a \$10 initial bet on this hand is \$4.05 with a double and \$2.60 with a hit.

Ace-seven versus a dealer's six-up is analogous. Basic strategy is to double down, although many players stand. The math shows that doubling yields greater expectations over repeated instances of this hand -- \$3.83 for a \$10 initial bet as opposed to \$2.80 by standing. However, success rate is higher for standing than doubling - 64 percent compared with 60 percent. And, over the short term, conservative bettors know they can not only lose more by doubling than standing, but hesitate to risk having their positions eroded by drawing four through nine.

Not every potential doubled hand offers the three-way trade-off among chance of success, expected profit, and exposure to loss. On hard 10 or 11 versus a dealer upcard of four through six, rational players would end up drawing only once regardless of whether they'd hit or doubled. So the probability of success is the same either way. Basic strategy prescribes doubling because expected profit is twice as high. There remains, though, the disincentive that exposure to loss is also twice as great.

A specific example involving equal chances of success is five-six versus six-up. Hitting or doubling, the win rate is 67 percent. But the expected profit for a \$10 initial bet in this situation is \$6.82 by doubling and \$3.41 by hitting. Another case might be three-seven versus a dealer's four. Either way, success rate is 62 percent. Expected profit on a \$10 initial bet on this hand is \$4.78 doubling and \$2.39 hitting.

If you're wondering, the converse condition doesn't present a similar opportunity. When basic strategy is to hit, not double, chance of success and expectation are both better by the book. Take five-six versus ace. Excluding the logical "don't care" of a dealer blackjack, success rate is 57 percent hitting and 53 percent doubling, and expected profits on \$10 initial bets are \$1.48 and \$1.29, respectively. So hitting is better by both criteria and involves less exposure to loss as well.

These gambles illustrate conflicts between probability and decision theories. What should you do? Idealists say play the percentages by maximizing expected win - the laws of probability. Utilitarians say evaluate personal response to success rate and exposure to loss as well as expectation - decision or game theory. And, in trying to propitiate such ponderous predicaments through poetry, the celebrated Sumner A Ingmark says: