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Best of Alan Krigman

26 March 2012

Some casino games offer bettors opportunities to select wagers from along a spectrum between those in which each decision has a strong probability of winning a relatively small amount, and a remote chance of making a fairly substantial score. Few folks consider casino gambling to be a one-shot affair, though. Most think in terms of sessions. They hope their stakes will carry them through the inevitable normal downswings of fortune to strike it rich with a glorious stroke of fate in a jackpot-oriented situation, or to accumulate what they believe to be a worthwhile profit in a hot run of incremental payoffs. Inquiring minds accordingly want to know the implications that trade-offs on individual bets have on sessions as a whole. That is, how they affect the likelihood of a player lasting for a session of acceptable duration or of reaching a satisfactory earnings level, in either case before exhausting what they regard as an affordable budget.

The house advantage or edge on various bets influences the outcome of a session. For the statistically small number of trials typically undergone in particular games, however, the volatility of the wager has greater ramifications on performance prospects. The math mavens measure volatility using a parameter they call “standard deviation,” which can be simply envisioned as the representative bankroll jump up or down when a bet is resolved.

The impact of volatility can be shown by comparing several ways to risk \$12 per coup at double-zero roulette. For this purpose, say a player with a \$250 stake undergoes a session betting on one of the following six propositions, all having 5.26 percent house advantage, on every spin:

A. One spot: chance is 1/38 (2.63 percent) and payoff is \$420; standard deviation is \$69.15.
B. Two-spots: chance is 2/38 (5.26 percent) and payoff is \$204; standard deviation is \$48.23.
C. Three-spots: chance is 3/38 (7.89 percent)and payoff is \$132; standard deviation is \$38.83.
D. Four-spots: chance is 4/38 (10.53 percent) and payoff is \$96; standard deviation is \$33.14.
E. Six-spots: chance is 6/38 (15.79 percent) and payoff is \$60; standard deviation is \$26.25.
F. Twelve-spots: chance is 12/38 (31.58 percent) and payoff is \$24; standard deviation is \$16.73.

The laws of probability can be applied to ascertain the likelihood that solid citizens won’t exhaust their bankrolls before completing a specified number of coups. The accompanying table of survival probabilities gives the prospects of still being in action after 40, 80, and 120 spins – corresponding roughly to one, two, and three hours of roulette. The data show that bets with the highest hit rates and lowest payoffs give players the most promise of their stakes keeping them in action for sessions of any desired duration.

Probability of being in action with a \$250 stake for stated numbers of spins

```Bet                      40 spins   80 spins   120 spins

One spot                    41%        29%         23%
Two-spot split              56%        40%         32%
Three-spot row/street       66%        48%         38%
Four-spot corner            73%        54%         44%
Six-spot double row/street  84%        64%         52%
Twelve-spot dozen/column    97%        84%         71%```

An alternate application of the laws of probability can be utilized to determine the chance players will attain stated target profit levels before depleting their resources, notwithstanding how long it takes to flourish or falter. The accompanying table of profit probabilities shows the likelihood of success for earnings of \$125, \$250, and \$500. The data indicate that bets with the highest payoffs and lowest hit rates give players the brightest outlooks for achieving desired overall gains.

Probability of reaching stated earnings targets before exhausting a \$250 stake

```Bet                        win \$125   win \$250   win \$500

One spot                       66%        48%        31%
Two-spot split                 64%        47%        29%
Three-spot row/street          63%        45%        27%
Four-spot corner               62%        43%        24%
Six-spot double row/street     59%        39%        20%
Twelve-spot dozen/column       47%        24%         7%```

Deep down inside, most casino aficionados would like to maximize their chances on both counts, if they could. Long sessions for the entertainment value and more chances to win, and high earnings for the benefits money can buy. Like many things in life, gambling isn’t like that. It generally involves optimization rather than maximization – seeking to balance elements that tend to work against one another. A strategy offering chances high enough along the dimension of primary importance, yet not unacceptably low along any other. No single point of compromise is ideal for everyone. The trick is for players to recognize the effect of the trade-offs and utilize that knowledge to devise a best plan of attack for themselves. The pragmatic poet, Sumner A Ingmark, captured the concept with this injunction:

Set objectives practically,
Then approach them tactically.