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# Numbers Can Fool You, in or out of the Casino

27 November 1995

Statistics can be misleading. Often inadvertently, sometimes deliberately. Casino gambling, based as it is on numbers, is especially vulnerable. But errors in statistical reasoning also plague the "real world" where their impact is far more serious.

Casino hoopla giving totals paid by slot machines during some period are an example. The figures don't tell what went into the machines to get the indicated sums out, so they're meaningless. Fanfare flaunting giant jackpots won by the fortunate few are another illustration. They're hollow because they don't relate amounts to likelihood of winning. But deceit's not necessarily involved. These numbers are just part of the razzle-dazzle that makes casinos exciting for the solid citizens dashing through the doors.

I'll show you how easy it is to draw false conclusions from true statistics. I'll use a mistake that's common in interpreting data: treating precise concepts in an imprecise manner.

Say I claim to have a system that makes the chance of winning one and a half times better in single-zero than in standard double-zero roulette. If this were true, and if it meant what it sounds like, you'd be nuts to play at a double-zero table when single-zero wheels aren't all that tough to find.

Here's the system. Bet \$1 on each of 35 individual numbers. If one of the numbers hits, you net \$1. If none hits, you lose \$35. A buck is a small win for \$35 at risk, but you're strongly favored.

At a double-zero table, the wheel has 38 positions. You've got 35 covered and only three open. So the odds are 35-to-3 or 11.67-to-1 that you'll win.

At a single-zero table, the wheel has 37 positions. You've got 35 covered and only two open. So the odds are 35-to-2 or 17.5-to-1 that you'll win.

Your chances of a hit are good in either case. But 17.5 is one and a half times greater than 11.67, so the single-zero game is 1.5 times 150 percent better than the double-zero version.

This "logic" plays loose with the lingo. It's more valid to compare probabilities than odds. In double-zero roulette, the probability you'll net \$1 with \$1 bets on each of 35 numbers is 35/38 or 92.1 percent. In single-zero games, the probability is 35/37 or 94.6 percent. Single-zero still looks better, but a win is an incremental 2.5 percent more probable, not a huge 1.5 times likelier.

The United States Supreme Court made essentially the same mistake in McClesky v Kemp. The Court accepted evidence in this case showing "defendants charged with killing white victims were 4.3 times as likely to receive a death sentence as defendants charged with killing blacks." According to Arnold Barnett of MIT, the statistical study cited by the Court did conclude "the odds of a death sentence in a white-victim case were 4.3 times the odds in a black-victim case." But this didn't mean that a death sentence was 430 percent more likely in the one than the other.

Professor Barnett explained that if the probability of a death sentence in an egregious white-victim homicide were a near-certain 99/100 or 99 percent, the associated odds would be 99-to-1. If the odds of a death sentence in a comparably heinous black-victim murder were cut by a factor of 4.3, they'd be 23-to-1. But, 23-to-1 odds corresponds to a probability of 23/24 or 95.8 percent, still almost a certainty. The more precise data interpretation also suggests discrimination, but a difference of 3.2 percent is much less alarming than a factor of 4.3.

Again you have it! Games imitating life. And a lesson, whether you play for fun or for keeps, that it may be a poor idea to make a firm commitment based mainly on numbers you heard on TV. Or read in a book. Or even saw in an authoritative newspaper column like this. As Sumner A Ingmark, the statistician's storyteller, stated so strongly:

Mathematical confusions
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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.