Hold Percentage
Gambling Ramblings: Extra Stuff by P. Griffin

Mathematicians frequently confuse, and are confused by, casino personnel in discussions of win rate (or %) for a game like Roulette. The mathematician describes the house advantage as 5.26% for a color bet in Roulette because for every 38 units wagered the house expected to win a net of 20-18 = 2 units since there are 20 ways for it to win and only 18 ways for the player 2/38 = .0526 = 5.26% is then the expected gain per bet made and is the "mathematical percentage."

Casino bosses have, however, evolved a different method of describing the performance of, for instance, a Roulette table. They might say that "The PC in Roulette is 20%" or "The Roulette table holds 20%." This figure is an empirical one necessitated by the type of bookkeeping casinos use to monitor performance and appears greatly at variance with the mathematicians' 5%. Who's correct?

Well, a better question is "How is the 20% hold figure arrived at in Roulette?" To illustrate the casino point of view we'll start with the simplest possible example: one Roulette table and only one gambler. Suppose the player buys in for $100 worth of chips and plays for exactly one hour. He may occasionally be ahead or possibly be wiped out within that hour, but let's imagine he terminates his play (either of 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.
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