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Best of Alan Krigman

Gaming Guru

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Why Your Gambling Goals May Defy What the Math Says To Expect.

16 November 2005

In baccarat, considering only rounds that don't tie, the probabilities of winning are 49.3 percent for Player and 50.7 percent for Banker. This is sometimes expressed as "Player should (or is expected to) win 493 out of every thousand hands, and Banker should (or is expected to) win 507 out of a thousand."

But don't expect what the math says you should. Pretend you sit through a thousand decisions, about four hours of minibaccarat, betting $10 per round on Player. Winning the "expected" 493 hands would put you $5,070 - $4,930 or $140 in the hole. This deficit results from the house edge. However, nobody would play $10 baccarat for four hours expecting to lose $140. The ego trip of a comp at the $13.95 all-you-can-eat buffet isn't worth that much.

In fact, solid citizens who enjoy the action and use an exit strategy such as "four hours or bust" have reason to expect they'll win at least half their rounds, 500 or more in this case. That is, to anticipate breaking even or coming out ahead. The disparity arises because "expected" has a different denotation in ordinary parlance than in statistics. There, it refers to an average. Under many circumstances, it's also the condition with the largest chance of occurring, although usually not by much.

For instance, in a thousand bets on Player, the chance of exactly 493 wins is 2.523 percent. That of being one win away, 492 or 494, is close at 2.518 percent each. Similarly it's 2.503 percent each for 491 or 495 wins and 2.478 percent for 490 or 496.

More useful to know is the likelihood of hitting or exceeding the break-even 500-win mark. The chance of a wash after a thousand decisions, 500 wins and losses, is 2.287 percent close to the expected value. The probability of a $20 profit, winning 501 and losing 499 hands at $10 each, is 2.219 percent. A $40 profit, 502 successful rounds, is 2.145 percent. Chances fall faster as potential profit rises further. For a $100 gain (505 wins), it's 1.891 percent. A $200 net (510 wins) is 1.415 percent. And the probability drops below 1 percent at $300 in earnings (515 wins).

While the percentages for particular small wins or losses are comparable in magnitude, the line between agony and ecstasy becomes sharply drawn as sessions proceed. Overall, the chance of completing a thousand hands where you started or in the money, by winning at least 500, is only 34 percent. The complementary likelihood of losing $20 or more is nearly double, at 66 percent.

And that 66 percent includes losing a lot more. Make believe you have ample resources, so normal downswings don't send you to the lockers. The flip side of that chance of breaking even or winning is 34 percent of obtaining 486 or fewer wins, a loss over $260.

Your potential for profit improves as games get shorter. Perhaps you gamble by the clock but only want half as much time two hours, about 500 rounds. The chance of finishing even or with a profit improves a bit. It's 39 as opposed to 34 percent.

In the limit of one round, your chance of joy is 49.3 percent. At three rounds, it's 49.0 percent. This drops to 48.7, 48.5, and 48.3 percent for five, seven, and nine rounds respectively. Keep plugging away for 25 rounds and it falls to 47.2 percent.

Prospects are brighter if you'd be happy breaking even as well as finishing flush. For this, you need even numbers of rounds with equal numbers of wins and losses. The probability then starts at 74.3 percent for two rounds, 67.7 percent for four, and 64.3 percent for six. It's down to 55.3 percent for 24 rounds.

The underlying gambling principle is that you can establish multiple goals. These might include playing time on your stake, and a profit with which you'll be satisfied to quit. But the two tend to counteract each another. The higher you set the one, the lower your likelihood of achieving the other. This doesn't mean it's an "either-or" proposition; rather, recognize it as a compromise. Like most choices in and out of the casino. Here's what the wordster, Sumner A Ingmark, thought about the enigma:

An optimized trade-off affords you the best of it,
Establishing balance is truly the test of it,
The devil and all leave you only the rest of it.

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.