Why Casinos Like the Long Run, and Players the Short

8 June 2005

By Alan Krigman

Most gamblers realize that a probability such as one out of six doesn't mean a hit is in the bag after five misses in a row. It implies that over many instances, a span often glibly referred to as "the long run," hits are expected on close to one sixth of all trials. For instance, approximately six million occurrences of seven for every 36 million throws of the dice. Dividing 6 million by 36 million gives an average of 1/6 or about 16.67 percent.

Casinos rely for earnings on the edge or house advantage being applied to enough betting decisions that tallies of outcomes are close to the theoretical averages. The opposite for bettors. Players depend for profits on sufficiently small numbers that departures from the law of averages prevail and produce more than the projected number of wins. Of course, the converse can happen during short time spans, too. Frequencies below the anticipated averages are possible, and may exhaust a bankroll more rapidly than a literal interpretation of probability would suggest.

Nothing about statistics forces results into line in the long run, however. For example, make believe a craps aficionado sees one seven on 31 consecutive rolls. To average one out of six on 36 tosses, the next five hurls must all be sevens. Assuming nobody's played hanky-panky with the hexahedrons, the chance of a seven is still one out of six each time.

To envision how the law of averages does work, suppose only a single seven appeared in 36 rolls five less than expected. The average is one out of 36, and the fraction 1/36 equals 2.78 percent. A far cry from the mathematicians' 16.67 percent.

Say the dice are thrown 324 more times. Here, 54 sevens and 270 other numbers are expected. The randomness responsible for the dearth of sevens in the first 36 rolls could now yield an excess, bringing the total for the combined 360 rolls to the magical 60. But the law of averages doesn't hinge on such serendipity.

Indeed, sevens may also be wanting in the new 324 throws, perhaps 50 rather than 54. The total for the combined 36 + 324 or 360 rolls is 1 + 50 or 51 sevens. The average is 51/360 or 14.17 percent. The sevens "missing" from the first 36 rolls haven't been "made up." The gap has grown. But the average has gone from 2.78 to 14.17 percent, much closer to the expected 16.67 percent.

Go further. Maybe another 3,240 rolls. A sixth of these, 540, are expected to be sevens. What if only 530 occurred? The deficit rose by 10 to 19. The average, however, is 581 sevens divided by 3,600 rolls. This equals 16.14 percent. Closer yet to the predicted average, despite the sevens continuing to be elusive.

At this point, inquiring minds will want to know just how many rounds are needed to get to the long run where the averages hold. Alas, there's no simple answer to this key question. Statistical analysis offers a way to calculate the chance of being within any given range of the expected value after this or that many trials. But the full impact of the effect might better be illustrated by a computer simulation indicating the number of rounds needed before the actual frequency of an event gets to the theoretical average and stays there through at least 25 more successes.

Such a simulation not only suggests how many rounds it takes, but shows the enigma of how variable the number can be. In one set of 10 tests, with an expected average of one out of six, the actual frequency converged on 16.7 percent after as few as 4,129 trials. The next lowest numbers of trials were 6,612 and 10,289. One of the 10 trials failed to settle at the average after 100,000 attempts. Another required 75,930, followed by 54,945 and 39,203.

Which explains why there are winners and losers among the solid citizens, while bosses who keep their games jumping needn't worry about the bottom line. Unless, of course, they have thousands of nickel-dime players and one or two really high rollers. And, if you think about the averages, you'll understand exactly why. Here's how the inkster, Sumner A Ingmark, interpreted this idea:

While many perturbations are statistically concealed,
One oversized phenomenon is readily revealed.