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Why you can't judge a slot machine by its belly glass

19 December 2011

Gambling at a slot machine is a venture into terra incognita. The only map you have is a list of amounts paid for the various winning combinations, as delineated on what’s traditionally called the belly glass. You get no clues about the associated chances. And, absent this probability information, you can’t determine the key statistical parameters of the game and reliably estimate your prospects for a session or compare alternate devices.

Pretend a casino has two machines positioned side-by-side, each with the same return schedule, as shown in the first column of accompanying table. The games may look identical but differ in the chances of hitting the corresponding payouts. The odds indicated – which you’d have no way to know – are typical, but only two of a virtually limitless number of possible configurations.

Payout schedule with typical odds for two “look-alike” slot machines
```return  game 1 odds      game 2 odds
2500    262,143.0-to-1   262,143.0-to-1
1000    87,380.3-to-1    65,535.0-to-1
200     6,552.6-to-1     5,241.9-to-1
150     5,241.9-to-1     2,620.4-to-1
100     2,620.4-to-1     1,047.6-to-1
50      1,746.6-to-1     654.4-to-1
30      1,309.7-to-1     435.9-to-1
25      1,047.6-to-1     373.5-to-1
20      654.4-to-1       326.7-to-1
15      581.5-to-1       261.1-to-1
10      36.4-to-1        130.1-to-1
4       31.8-to-1        86.4-to-1
3       28.1-to-1        51.4-to-1
2       25.2-to-1        36.4-to-1
1       7.0-to-1         7.0-to-1
```

Most casino honchos know the edge – or its complement, the return percentage – on their slots. But few care about the constituent probabilities the machine vendors set internally at the various levels. Edge is crucial to the establishments because it governs how much of the cash flow they can expect to keep over an extended period. On the slots depicted in the table, these fractions work out to 5.7 percent (94.3 percent return) for game 1 and 9.7 percent (90.3 percent return) for game 2. If \$1 million passes through each of the machines, the casino should accordingly retain roughly \$57,000 from the first and \$97,000 from the second. The predicted earnings reflect the net of players who lose more than the edge would anticipate, lose less, or win.

Although slot aficionados are affected by edge, conventional wisdom aside, few gamble enough for this feature to dominate their fortunes. What most solid citizens consider good or bad luck follows from the volatility of the games, the bankroll jumps caused by successes and failures. Volatility is gauged by a quantity the math mavens call “standard deviation.” One decision on an even-money proposition with no pushes would have a standard deviation of \$1 per dollar bet, this being what’s added on a win or subtracted on a loss. In actuality, edge reduces standard deviation, but almost negligibly. For the hypothetical slot machines in the table, standard deviations are \$7.49 for game 1 and \$8.66 for game 2. The round-by-round primacy of edge over volatility can be seen by comparing 5.7 or 9.7 cents, the theoretical losses due to edge on a dollar bet, with \$7.49 or \$8.66, the standard deviations, on the respective machines.

Edge moves money one way, from bettors to bosses, and has a cumulative effect as the action progresses. Volatility is bidirectional, increasing as well as depleting a bankroll, with ups and downs tending to be mutually cancelling given enough decisions. So, the impact of volatility ultimately zeroes out and leaves only that of aggregated edge. Volatility therefore explains why individuals can win despite the house advantage, and also why they can lose far more in the short haul than the product of the edge multiplied by their gross wagers would suggest.

Players’ prospects of profits and of surviving the downswings of a game on a given bankroll before going broke can be calculated from the values of edge and standard deviation. For instance, on the machines cited as examples, the chance of earning \$500 before scraping the bottom of a \$100 fanny pack is 10.0 percent on game 1 and 7.9 percent on game 2. The likelihood of an initial \$100 poke sufficing to last for at least four hours at about one spin every 10 seconds is 20.6 percent on game 1 and 15.0 percent on game 2.

So, does it matter whether patrons know enough to choose rationally between these – or any other – games? When the dust settles on a million folks with \$100 bankrolls, 100,000 will have earned \$500 or more before losing their stakes on game 1, as opposed to 79,000 on game 2. Similarly, after four hours, 206,000 will be in action on game 1 and 150,000 on game 2. But, say a particular punter makes \$500, or is still in the running at the end of four hours. Who’s to say this person was among the bigger bevy of bettors on the looser than the tighter machine? In the course of a few hours or even several casino visits, nobody can distinguish between the two groups. In the words of the wily warbler of wagering, Sumner A Ingmark:

When fate on randomness depends,
What probability portends,
Reality quite oft transcends.