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Best of Alan Krigman

Gaming Guru

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Why Luck Goes beyond the Law of Averages

23 March 1998

Businesses such as banks and supermarkets can optimize their operations by balancing customer wait times against staffing needs. The starting point is average customer load during various periods. But savvy managers go beyond the law of averages. They employ a statistical tool, the Poisson Distribution, to estimate the chances of encountering heavier or lighter demand.

Casino players can also use this concept to go beyond the law of averages. Doing so can help them understand luck - departures from "expected" results, why one machine or table seems hotter or colder than another despite identical odds and edges.

Say a solid citizen is sure casino bosses set the slot machines by pushing buttons somewhere. The player "charts" several casinos on busy nights, counting how often jackpot bells ring during 10-minute intervals. The person will then play where the number is highest, thinking it's where the slots are currently the loosest.

Make some assumptions for the sake of argument. 1) Each charted casino has 2,000 machines in action during the observations. 2) Players average four spins per minute. 3) Odds and payouts are the same everywhere, with jackpots being 40,000-to-1 longshots.

Under these conditions, an average of two jackpots can be expected in any 10-minute interval. But what are the probabilities of fewer, exacty two, or more jackpots during such a period? Answers are provided by the Poisson Distribution. They are 13.5 percent for none, 27.1 percent for one, 27.1 percent for two, 18.0 percent for three, and 12.6 percent for four or five. As a result, during the 10-minute checks, a player is about as likely to find no jackpots at one casino and four or five at another - even though edge and chances of a hit are the same.

And, what if the machines in one casino differed from those in the rest? Suppose the odds against a jackpot at one establishment were 50,000-to-1. The average there drops to 1.6 hits every 10 minutes; actual chances are 20.2 percent for none, 32.3 percent for one, 25.8 percent for two, 13.8 percent for three, and 7.2 percent for four or five. Down from the houses with 40,000-to-1 machines, but not by enough so that counting the number of bell-ringers in a 10-minute interval yields any useful information.

Similar statistics govern other games. For instance, the probability that a blackjack dealer will "break" is roughly 28 percent. In an 18-round shoe, the dealer can be expected to break five times. But, what are the chances of a hot shoe in which the dealer breaks eight or nine times? Or of a cold shoe in which the dealer only breaks once or twice - or not at all? The answers, again provided by the Poisson Distribution, are as follows:

# of breaks
probability (%)
0
0.7
1
3.4
2
8.4
3
14.0
4
17.5
5
17.5
6
14.6
7
10.4
8
6.5
9
3.6
10
1.8
11
0.8
12
0.3

 

Here are examples of how to interpret these figures. The average number of dealer breaks during an 18-round shoe is five. However, the chance of exactly five breaks is only 17.5 percent. Moreover, there's a 1.8 percent chance the dealer will break 10 times - twice the expected rate. Likewise, there's a 3.4 percent chance the dealer will only break once during the shoe. And a small but real 0.7 percent chance the dealer won't break at all.

Next time you're at BargainMart and the cashiers leave for lunch just as shoppers converge on the checkout aisles, consider this. Did you hit a normal but low-probability event, or did management ignore statistics in planning? And next time you're gambling and hit a hot streak, think about this. Are you experiencing the extremes of the Poisson Distribution, or did your 1-900-PSYCHIC advisor really know it would be your lucky day? Either way, remember the reasoning of that rational rhymer, Sumner A Ingmark:


When minor setbacks get you scared,
The fault may be you're unprepared.

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.