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# Playing it smart - What every gambler should know about progressions

14 May 2007

Progressions are important to gamblers. In betting strategies. And in figuring the probabilities of winning or losing sequential or combined wagers. Progressions are also key elements in fields as diverse as economics, engineering, biology, and epidemiology. Which goes to prove once again, as if proof were needed, that everything you need to know in life you can learn in a casino.

The math mavens earn those big bucks by labeling progressions with fancy names so they seem more complicated than they are. The major kinds are called arithmetic (pronounced with the emphasis on "met") and geometric. You can think of arithmetic progressions as series which get from one stage to the next by adding or subtracting. Generally a constant amount 5, 10, 15, 20, and so forth. Geometric progressions go up or down by multiplying or dividing. Typically by a fixed factor 5, 25, 125, 625, and on.

Without getting too deep in the nitty-gritty, you can see that the major effective difference between the two categories is that geometric progressions change a lot faster than arithmetic series. In six steps, for instance, adding \$5 to a starting bet of \$5 gets you to \$35. Multiplying a \$2 bet by two in six steps gets you to \$128.

Solid citizens progress bets for one of three main reasons. To reduce wagers during a game and lock up profits, to increase bets when they believe they're on a tear in hopes of reaping big returns by reinvesting money they didn't have when they started but picked up along the way, and to recoup earlier losses.

The wisdom or folly of progressive betting for any such reasons is usually evaluated with 20/20 hindsight. It would be preferable to plan by considering the implications in advance. For example, say you start an even-money game like blackjack betting \$5 and won't be happy unless your \$100 poke becomes \$1,000. You could do it with a geometric progression, but should realize you're headed for a short stint more apt to end in the hole than in the clover. Conversely, if you're ahead at craps during a long roll and neither want to take everything down while numbers keep popping nor leave too much on the table when a seven eventually shows, a subtractive arithmetic progression might be the ticket.

On the probability side, sequential events operate geometrically while simultaneous incidents are arithmetic. This generally means that your chance of multiple hits in series drops sharply as the chain you need lengthens. And chance of success on groups of bets in single coups grows slowly as you cover more conditions.

Roulette offers illustrations of both phenomena. Say you want to bet \$1 on a single spot and make big bucks by parlaying it if you win. You're talking steep geometric progressions. A first success pays \$35. Leaving all \$36 on the number brings \$36 x 35 or \$1,260, for a net profit of \$1,295, on the second. The chance of hitting a single number at a double-zero table is 1/38 (2.63 percent); that of two in succession is (1/38) x (1/38) or 1/1,444 (0.069 percent). Three in a row would net \$45,655 but the chance falls to (1/38) x (1/38) x (1/38) or 1/54,872 (0.0018 percent).

Conversely, maybe you decide to take a shot with \$10 on a single round. Bet it all on one spot and you have that 1/38 chance (2.63 percent) to win \$350. Bet \$5 on each of two spots and you're looking for \$170 profit but your chance improves to (1/38) + (1/38) or 2/38 (5.26 percent). At \$2 on each of five spots, earnings are down to \$62 with a likelihood of (1/38) + (1/38) + (1/38) + (1/38) + (1/38) or 5/38 (13.16 percent). Earnings decline and probability rises in gradual arithmetic progressions.

In games of independent trials, such as roulette, your chances on one round are the same as on any other. So, if you just bested the 1/38 probability of winning on a single number, you're facing the same prospects on your next wager. But this isn't the same as committing to go for broke on a parlay or sacrificing potential profit by splitting your total at risk between multiple spots. A realization thus remarked by the rhymer, Sumner A. Ingmark:

Be wary of sessions involving progressions,
They feed on impressions that prompt indiscretions.

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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.