Should you bet on the trend you perceive or did just it occur by chance?

Casino aficionados often envision trends in the outcomes of games. From general phenomena such as machines or tables being hot or cold to specifics like black numbers dominating roulette hits, craps throws landing especially frequently on fives and nines, or video poker hands building toward a jackpot. Unless the game is biased in some way – for instance, owing to unbalanced roulette wheels, loaded dice, or faulty computer chips – the observed conditions occur by chance.

There’s a way to help distinguish real tendencies from pure coincidence. A statistical procedure called a “chi-square test for goodness of fit” can be used to indicate the likelihood that a series of events reflects a genuine bias as opposed to mere chance.

To illustrate, make believe you scout a double-zero roulette pit looking for a trend to exploit. At one table, you notice that black numbers hit more often than red. To convince yourself, you clock 38 spins. If the observed results precisely tracked the probabilities inherent in the game, under the assumption that outcomes are random, you’d find 18 reds, 18 blacks, and two greens. Instead, you see 16 reds, 21 blacks, and one green. Were the wheel to be somehow off-kilter or the dealer to handle the ball in such a manner that these results were repeatable, bets on black would give you an advantage over the house. You’d have 11.8% or 10.5% edge, depending on how the bosses resolve outside bets when the ball lands in green (player loses half or player loses all). At $100 per round, on the average for every 38 spins, you’d expect to lose either $1,650 or $1,700 and win $2,100 for a net $450 or $400 profit, respectively.

The chi-square test can be invoked to compare the expected frequencies with the hypothetical observations cited. It indicates that the data for the 38 spins would have 45.7 percent or less probability of being obtained by circumstances other than pure chance. Not a bona fide trend.

Imagine, instead, that you recorded the results of 190 spins. With an unbiased wheel, you’d expect 90 reds, 90 blacks, and 10 greens. You got 80 reds, 105 blacks, and 5 greens. The proportions are the same as for the 38-spin example, but you have fivefold the data. The 190 values yield a chi-square probability indicating up to 95.3 percent probability that the phenomenon is systematic and not a fluke. An even longer run showing the identical proportions would afford a yet higher degree of confidence that you’ve uncovered a non-random situation. After 380 spins, 160 red, 210 black, and 10 green rather than the expected 180 red, 180 black, and 20 green would have a chi-square probability as great as 99.8 percent of occurring by design or structure rather than chance. For the same relative departures from the unbiased expectations, the more extensive the data, the higher confidence you can place in drawing your conclusions.

Chi-square tests can also highlight differences in the probability that weak and strong departures from expected values would occur by chance. Say you think you’ve deduced a way to set and throw dice at craps to lower the frequency of sevens. Your idea is to hold one die with your thumb and forefinger on the 3 and 4 and throw it so it rotates like a wheel around the 3-4 “axle;” ideally, these numbers would never roll. Your system also requires you to do something similar with the other die but with other faces on the ends of the axle. To check the feasibility of the concept, you try the 3-4 plan with one die. In 60 completely random throws, you’d expect each face to appear 10 times. If you use your technique perfectly on 60 throws, you’d get no 3's or 4's and 15 each of 1's, 2's, 5's, and 6's. A chi-square test shows that such a radical shift from the values expected with random throws yields over 99.9 percent confidence in your technique.

Wizened craps buffs know, however, that many things can interfere with and influence the dice before they come to rest. And your system will give you an edge if you cut back on but don’t completely eliminate 3's and 4's, so your throws needn’t be perfect. You check your skill with 60 throws and get four 3's and four 4's along with 14 1's, 13 2's, 12 5's, and 13 6's. A chi-square analysis shows that the probability of this result other than by chance approaches 94.9 percent, pretty much verifying your setting and throwing skills. A friend tries your technique and gets eight 3's, seven 4's 13 1's, 11 2's, 9 5's, and 12 6's. The 3's and 4's are rare enough so you could make money if this distribution was consistently achievable. But a chi-square analysis indicates it could happen by chance with a probability of at least 73.1 percent, so you’d have no more than 26.9 confidence it reflected your pal’s ability to work the system successfully over the long haul.

Trends casino players often think they observe are usually, but not always, about as real as the bear in Ursa Major or the lion in Leo. But you needn’t simply guess. You can calculate the statistical parameters needed to substantiate your fantasy. Unfortunately, doing so requires more data or evidence of especially strong biases than most solid citizens are willing to gather before taking their shot. So they’d be wise to heed this warning by the warbler, Sumner A Ingmark:
A gamblers’ theories don’t much matter,
Unless confirmed by lots of data.