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Probability and order in gambling

8 November 2010

In many casino games, it's relatively simple to enumerate all possible outcomes of a coup and identify their associated probabilities. Such listings yield statistically-correct theoretical "cycles."

Double-zero roulette offers a readily-visualized illustration. The ball can finish in one of 38 equally likely grooves. Players accordingly "expect" three wins and 35 losses per 38 spins when betting on any three-number row. "Expect" has a special meaning in this context. A row hitting twice in 37 spins doesn't foretell a third hit on the 38th try. Rather, it indicates averages approached over huge numbers of trials. Expectation is accordingly descriptive, not predictive, and is mostly useful in determining factors such as the prospect of winning on any spin and the edge the bosses have over players. For the roulette example, the chance of winning is three out of 38. And, knowing the casino pays 11-to-1 on a row, the house edge is (3/38) x 11 - (35/38) x 1, which equals 5.26 percent. The principle is identical in other games although theoretical cycles can get so large in some cases that the enumerations require complex computer calculations.

If you play through the number of coups represented by a theoretical cycle, you don't want the results to be statistically correct.. If they are, you'll lose precisely the amount determined by multiplying gross wager times house edge. Consider a $1 bet on a three number row at double-zero roulette. In a statistically-correct 38-spin run, you'd win 3 x $11 and lose 35 x $1 – a net $2 setback. And a $38 gross wager multiplied by 5.26 percent edge equals $2. You'd like your short-term results to differ from the long-run averages. The catch is that you can't anticipate when departures from the average will help or hurt. Four hits would gain you $44 and the remaining 34 misses would subtract $34, leaving you $10 profit; additional hits would be even more desirable. Two hits would leave you behind by $22 - $36 or $14; fewer will be costlier yet.

There's something beyond probabilities to think about. The order in which the events occur.

Again, pretend you're betting $1 on rows at double-zero roulette. But, instead of always betting on the same row, you follow your hunches to choose a row for your current bet based on the results of previous spins. Now, even if the series is statistically correct, you might win money or give up more than the $2 in 38 spins. Your fate would depend on the order in which the numbers occurred and how well or badly this coincided with the rows you picked.

Comparable reasoning applies to Place bets in craps. Make believe you had $5 on the four for every throw of the dice. A statistically correct cycle of 36 throws would yield three fours, six sevens, and 27 numbers that had no effect. The bet pays 9-to-5. You'd pick up 3 x $9 or $27 and drop 6 x $5 or $30 for a net shortfall of $3. In craps, edge is determined only from rounds where there's a decision. That's three plus six or nine rounds betting $5 each, a $45 gross wager. The edge on the four is 6.67 percent. Take 6.67 percent of $45 and you get $3. Departure from the statistically-correct 2-to-1 ratio of sevens-to-fours could help or hurt, depending on the way the difference went. And order would matter if you kept switching the number on which you bet, causing you to win or lose regardless of whether or not the 36 throws were statistically-correct.

Solid citizens who consistently bet Pass or Don't Pass at craps are also affected by the order as well as the frequency of the various totals.. Cycles for these bets are 1,980 as opposed to 36 throws because they cascade wins, losses, and points established during come-out rolls with wins and losses on each of the possible points. However, the importance of order can be seen by contrasting the effects of the following two 18-throw segments. Each set of 18 contains the identical distribution of the totals – three 4s, four 5s, five 6s, and six 7s. But flat $5 Pass bets of would win $55 with the first sequence and lose $30 on the second. Conversely for Don't Pass..

7, 7, 4-4, 7, 5-4-5, 7, 6-5-6, 7, 6-6, 7, 6-6
4-6-7, 4-5-6-7, 5-4-6-7, 5-7, 5-6-7, 6-7

The phenomenon also occurs in card games. There, probabilities are set by the composition of the deck or shoe, but shuffling changes the order. Picture a blackjack buff head-to-head against the dealer. The player would win when the next four cards drawn were 10-10-A-10, and lose when the same cards came out as 10-A-10-10.

The laws of probability deal indirectly with order by means of volatility, which quantifies how swings in fortunes occur during the course of the action. However, seeking a way to quantify order as a means of overcoming edge is doomed to failure and confuses gambling with schemes to earn a living by beating the house. Or, as that prolific poet, Sumner A Ingmark, penned:
While measuring chance has undoubted allure,
It doesn't convert the uncertain to sure.

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.