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The law of large numbers and the anarchy of small samples24 September 2012
By Alan Krigman
The statistical “law of large numbers” tells analysts their degree of confidence in anticipating the collective outcomes of many operations whose individual probabilities are known. Anarchy reigns, however, for the combined results of relatively few transactions. Which is why well managed casinos can foresee how they’ll do over reasonable accounting periods while surprise awaits particular players during specific forays into the wonderful world of wagering.
Make believe a casino logs 10 million decisions on bets of $1 each, in a game with a 5 percent edge. The math mavens tell the bosses that, theoretically, they’ll earn 5 percent of the gross $10 million wagered – $500,000. Here’s how this works. Say players can make one of two wagers on this game. The first pays 1-to-1; the house gets 5 percent if players have 47.5 percent chance of winning versus 52.5 percent of losing. The second pays 9-to-1; the house gets 5 percent if players have 9.5 percent probability of success and 90.5 percent of failure. Were the results of the action statistically correct, 10 million decisions on the former would pay out $4,750,000 and take in $5,250,000, netting the house $500,000. Similarly, 10 million decisions on the latter would give players $8,550,000 and take $9,050,000, also leaving the casino with $500,000.
But wait, there’s more! A 5 percent edge doesn’t mean the house pockets a nickel whenever $1 is put at risk. Further, in practice, a tally of outcomes isn’t apt to be right on the statistically correct button. Still, as long as the law of large numbers holds, the bosses can be confident their earnings will be close to the value predicted by edge, and also know how confident and how close.
Based on numbers of decisions involved, and the unique sets of payoffs and probabilities that yield the theoretical edge, a range of outcomes can be calculated within which the bosses can have any specified degree of assurance they’ll fall. To illustrate, say $1 is bet 10 million times in the hypothetical 5 percent game. Were the bets all made at even money, the powers-that-be could have roughly 90 percent confidence of earning between $495,000 and $505,000. Had the $10 million been bet $1 at a time but on the wager with 9-to-1 payoff, the 90 percent confidence level would be from $485,000 to $515,000 in earnings.
With fewer decisions, the range of outcomes for any degree of confidence narrows in absolute value but broadens relative to the theoretically expected result. Say a nervous Nellie in the head office was worried about a table that often lost money during a certain shift. After 1,000 $1 bets, expected net would be 5 percent of $1,000 or $50. With 1-to-1 bets, chance would be 90 percent of finishing between $102 up and $2 down, and likelihood of ending in the hole would be over 5 percent. With 9-to-1 bets, chance would be 90 percent of finishing between a $202 ahead and $52 behind, and likelihood of ending somewhere in the red would exceed 29 percent.
As the number of trials decreases, the range for any degree of confidence gets wider and less meaningful. The situation is therefore highly uncertain for solid citizens, who undergo so few resolutions that random swings in their fortunes caused by round-to-round victories and defeats overwhelm the inevitable and certain erosive effect of the edge. As an example, pretend that Vinnie and Winnie each play the hypothetical game for 10 coups betting $1 per round. Vinnie bets on the 1-to-1 shot while Winnie goes for the 9-to-1 payoff. The accompanying table shows the profit or loss and the corresponding probability, with zero to 10 wins for either player.
Vinnie (1-to-1) Winnie (9-to-1) wins win/loss($) probability win/loss($) probability 0 -10 0.159071741% -10 36.854098483% 1 -8 1.439220513% 0 38.686622717% 2 -6 5.859683517% 10 18.274620123% 3 -4 14.137649120% 20 5.115547438% 4 -2 22.384611106% 30 0.939734543% 5 0 24.303292058% 40 0.118375401% 6 2 18.323910679% 50 0.010355123% 7 4 9.473586473% 60 0.000621144% 8 6 3.214252553% 70 0.000024451% 9 8 0.646251836% 80 0.000000570% 10 10 0.058470404% 90 0.000000006%
Multiplying the win or loss for the 10 coups by the corresponding probability for either player then adding the results yields an expected loss of $0.50, 5 percent of the $10 gross wager. Vinnie and Winnie can’t end their sessions losing $0.50, though. They can win or lose the amounts shown. And, if they’re like most players, they’d like to know their prospects for breaking even or emerging with a profit. Adding the probabilities of all such results shows that Vinnie’s chance is over 56 percent while Winnie’s is greater than 63 percent. Despite the house’s edge, in the short run, bettors have a decent shot at bringing home the bacon. And casino bigwigs responsible for hooking super-high rollers are vulnerable to being relegated to doghouse when those premium players win a few million bucks . As that awesome inkster, Sumner A Ingmark, observed: