Law of Averages
Introduction | The Law of Large Numbers |
A general idea that many people hold goes something like this:
The longer a series of chance events goes on, the more likely it is that things will "even up." For example, the longer you drive a car, the more likely you are to have your share of flat tires. The more children you have, the more likely you will have an equal division of boys and girls (assuming an even number of children). The longer you flip a coin, the more likely the number of heads and tails will equalize.
Actually, the latter two examples are identical if certain simple assumptions are made. In the case of the coin, we shall suppose that the coin is "fair", so that on any one flip, head and tail are equally likely. In the case of the children, we shall rule out twins or other multiple births, and shall assume that the parents do not have any tendency to produce children of one sex as preferred to the other. Of course there is the practical consideration that one can flip a coin a lot more times than a woman can bear a child. So, it is more natural to use the coin for purposes of illustrating large samples. But for small samples, there is something unique about each child, which makes the family a marvelous illustration for stressing certain points.
his own desire or because the casino closes the table) with $80 worth of chips, which he takes to the cashier to be redeemed in money. The pit boss analyses the table's performance by comparing the $100 drop in the cash box with the 80 missing chips in the money tray. The difference, 100-80 = 20, is the amount that this table "held" and the quotient of "hold" divided by "drop" or 20/100 = 20%, is the casino's hold percentage.
It's important to realize that the hold percentage is an empirical, even to some extent sociological, figure. It varies from day to day, year to year, and it especially varies with the characteristics of the players. The problem is greatly complicated when there are different ways to play a game such as Craps, which have different mathematical percentages and when players, as they frequently do, carry chips from table to table or game to game.
A couple of oversimplified examples may help to illustrate this dependence. First, suppose our previously described solitary gambler has enormous endurance and is determined to play forever or until he loses his entire $100 buy in. Since Roulette is an unfavorable game, this latter eventuality is the assured result. However long it takes, when our indefatigable gambler has finally lost all his money and the casino closes the tables, it will be discovered that all of the chips are still there and so too is the hundred dollar bill in the drop box. Hence the casino won 100% of the drop. A mathematician keeping score would count, perhaps, the 1900 dollar bets the gambler made and report the casino win rate as 100/1900 = 5.26%.
Now let's introduce another table and another gambler. Mr. A buys in for $100 at a table #1, plays for a while, and then walks away a net loser of $20. Mr. B buys in at table #2 for just one dollar, wins a dollar, and walks to the cashier a net winner. Now Mr. A walks over to table #2 with the 80 chips he bought at table #1. He plays for an hour and only loses two chips at table #2. How will the casino hold performance look for the two tables? As in the very first example, table #1 will show a 20/100 = 20% win rate. But what about table #2? Mr. B took away two chips, the one he bought and the one he won. Mr. A left two chips, the ones he lost. Hence the table has as many chips as it started with and concludes it "held" all the drop (Mr. B's drop incidentally, and he was a winner), or 100%.
As insusceptible as win/drop is to precise mathematical prediction it does nevertheless remain a useful method of description for casinos. The empirical percentages derive a long-term validity because of the large flow of action. Perhaps one exception to this is the game of Blackjack where education of the players has probably reduced the casino hold percentage over the years, although not the profits which continue to grow because of increased volume.
