Stay informed with the
Recent Articles
Best of Alan Krigman

Why faith in the law of averages is misplaced

5 September 2011

Gamblers enduring cold spells in their action often put their faith in the law of averages to restore balance and get them out of trouble rather than cut their losses and run. This faith is misplaced.

One reason is that averages – which may also be pictured as theoretically expected values – reliably indicate performance only for large numbers of samples. For fewer tries, well...

On the average, a blackjack is dealt about once every 21 hands. Deal 21 million hands and you’ll be within a reasonable margin of error of a million blackjacks. Deal 21 hands and your chances are 36.0 percent of none, 37.7 percent of one, 18.8 percent of two, 5.9 percent of three, and so on. One blackjack is the most likely result in the 21 hands, but it’s only slightly more probable than none.

A 14 (or whatever number you pick) pops at double-zero roulette an average of once every 38 rounds. Play 38 million coups and you’ll be within a reasonable margin of error of a million 14s. In 38 spins, your chances are 36.3 percent for none, 37.3 percent for one, 18.7 percent for two, 6.0 percent for three, 1.4 percent for four, and under 1 percent for five through 38. One in 38 is most likely but is only a bit more probable than none.

A seven appear an average of once every six trials when two dice are thrown. Toss the dice six million times and you’ll be within a reasonable margin of error of a million sevens. Hurl ‘em six times and your chances are 33.5 percent of none, 40.2 percent of one, 20.1 percent of two, 5.4 percent of three, and under one percent each of four through six. Again, one – the average – is most likely, but it’s not much more probably than none.

Sevens on dice can illustrate how chances approach the average as numbers of trials increase. For six throws, the probability of the theoretical average with one seven was 40.2 percent. For 600 throws, the chance of hitting the theoretical average with 100 sevens is 4.4 percent – far less, not more, than the 40.2 percent likelihood of being precisely on the button for six throws. The seeming contradiction is where the qualifying “within a reasonable margin of error” applies. The chance of being within 5 percent of 100 sevens, from 95 to 105 of them, in 600 throws is 45.3 percent. For 1,200 throws, the chance of being at exactly 200 sevens is 3.1 percent, but that of being within a 5 percent margin of error – between 190 and 210 sevens – grows to 59.4 percent.

House advantage or edge on every bet poses another problem for anyone banking on extended play to earn a profit from the law of averages. Results get closer to the average or expected value, such that swings in final bankrolls are moderated; however payoffs on bets are set so that, close to the average, players lose more from rounds they lose than they earn on those they win.

Pretend you bet \$1 on one number at double-zero roulette. If you play 38 rounds and hit once, the average, you’ll have lost \$37 and won \$35, finishing \$2 in the hole. You therefore have 36.3 percent chance of losing \$38 (no hits) and 37.3 percent chance of losing \$2 (one hit). That’s 73.6 percent chance of a loss and the other 26.4 percent chance of a profit. Say you play 380 spins and hit the 10 win average, you’ll have collected \$350 and dropped \$370 for a net deficit of \$20. You finish 380 spins with profits of \$16 at 11 hits, \$52 at 12, \$88 at 13, and so on. The overall chance of such good fortune is 41.7 percent. Losses at 10 or fewer hits have a 58.3 percent probability.

The order in which results occur adds another dimension to the fruitlessness of faith in the law of averages. In craps, for instance, a solid citizen betting on Pass or Come and experiencing the average number of sevens during a session could finish richer or poorer according to the extent to which this total occurred during come-out or point rolls, respectively. Alternately, in blackjack, a shoe has a fixed proportion of each rank, for instance four out of every 13 cards are 10s. A 13-card block of a shoe being dealt might happen to have four 10s. Whether they’ll help in a particular situation, for example when you’ve doubled down on an 11, isn’t a matter of the average but on how they’re arranged in that section. A four to you and a 10 to the next player is obviously not the same as the 10 to you and the four to the next person.

The ultimate error in trusting the law of averages is in believing there’s an underlying mechanism that maintains the equilibrium it represents. You’ve lost 10 Place bets in a row on the nine at craps when the law of averages leads you to think you ought to have won four? Should you press your bets on this number, anticipating eight wins in the next 10 decisions to bring the total up to eight out of 20 where you assume it belongs? There’s no driving influence of this type. Your chance is four out of 10 no matter what the proportions of your past wins or losses.

Perhaps we’d best stop here or we’ll find ourselves talking about factors that might make a game less random than either the bettors or the bosses – depending on the situation – want or prefer to believe it is. Because, as the inimitable inkmeister, Sumner A Ingmark, invidiously intimated:

Were there some special force or field, to which the laws of chance must yield,
Then those to whom it is revealed, a tool with power great would wield.