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# Can casinos earn more money than players put at risk?

28 June 2010

Make believe you go to a casino with a \$100 budget. You decide to risk it at your favorite slot machine. Although you can't know precisely, say the machine has a return rate of 92 percent. Here's what return rate means, in case you're unsure. Chances of wins at various levels and of losses are such that, in theory, the game should pay out 92 percent of the gross wager over many trials. The house's edge is the complement of return rate, 8 percent, the portion of the collective bet the joint should keep.

The accompanying table shows how return works on a hypothetical, simplified machine for a \$1 spin. You'd be aware of the payout levels. The associated probabilities would not normally be public information. Each of the expected values is the payout multiplied by the probability of its occurrence. The probabilities add up to 100 percent because one or another result will occur on each spin. The expected values sum to \$0.92, 92 percent of \$1.

Payouts, probabilities, and expected values for a hypothetical, simplified slot machine having a 92 percent return rate

```	payout	     probability	         expected value
\$1,000	       0.05%(1 out of 2,000)	    \$0.50
100	       0.15%(1 out of 667)	     0.15
1	      27.00%(1 out of 3.7)	     0.27
0	      72.80%(1 out of 1.4)	     0.00
overall	     100.00%		            \$0.92
```

Expected value is less complicated than folks typically imagine. Think of it like this. The expected value of a 25 percent chance of having \$1,000 equals 25 percent of \$1,000 or \$250. It's how much a 25 percent chance of \$1,000 is worth.

When you've finished playing, your bankroll could have shrunk to \$92. This would be coinci-dence, not a direct result of the 92 percent return rate. Experience shows you might also be even, ahead, or behind. In the latter situation, the loss might be your original \$100, less, or some greater amount you withdrew from a handy cash terminal. So, having an enquiring mind, you may want to know what the 92 and 8 percent have to do with anything.

Expected value, as shown in the table, is a statistical figure. It involves the chance rather than the result of any particular round. It's useful to bosses far more than to individual bettors because it anticipates how the machine performs over huge numbers of spins. If you bet \$1 in the hypo-thetical machine, you may win \$1,000 or \$100, break even, or lose \$1; you don't receive \$0.92.

On the other hand, the law of large numbers says that with enough trials, actual cumulative frequencies approach the probabilities programmed into the game. So, after a million \$1 bets, around 0.05 percent (500) will have returned \$1,000 -- \$500,000 in total, approximately 0.15 percent (1,500) will have paid \$100 -- \$150,000 in total, and roughly 27 percent (270,000) will have given back \$1 -- \$270,000 in total. In all, about \$920,000 will have gone to the solid citizens and \$80,000 to the casino. On the average, players will have been paid \$0.92 on the dollar.

People who think higher payback percentages mean easier to win aren't necessarily right. Pretend, for example, the hypothetical machine in the table topped out at \$2,000 and not \$1,000. The expected value of the jackpot would be 0.05 percent of \$2,000 or \$1.00. And the device would have 142 percent overall return. But the chance of winning either \$2,000 or \$100 would still be 0.20 percent, one out of five hundred, as in the 92 percent game.

Here's another point to ponder. Say you budget \$100 and play the machine in question for \$1 per spin. Your plan is to continue until you either go bust or hit the biggie. If you're fortunate enough to get one or two \$100 returns, but not so lucky as to grab the \$1,000, your intermediate earnings might suffice to keep trucking at something like 10 spins per minute for three hours. This is 1,800 spins at \$1 each, an aggregate wager of \$1,800. Theoretically, the casino will have returned 92 percent of \$1,800 (\$1,650) and kept 8 percent (\$144). So the bosses earned \$44 from you over and above the \$100 you had in your original stash. Is the casino making funny money paying itself a commission on its own funds, as the poet, Sumner A Ingmark, asked when he authored:

Do casinos earn profit from cash that they part with,