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Realistically, how often can slot players expect to hit jackpots?

16 July 2012

Chances of various final hands at video poker are no great secrets, assuming that players adhere to the optimal strategies. The theoretical frequency of trials that yield returns, at any particular level or on the whole, can be ascertained by anyone who cares to check into the matter. And, by multiplying the probabilities at each level by the associated payout then summing the products, the overall percentage return or its complement – the house’s edge – can also be determined.

On slot machines where the probabilities of success at various payout levels are arbitrary, the values are rarely made public. So, except in unusual cases where normally proprietary figures happen to be revealed, the statistical characteristics of the games are unknown. In a sense, they’re anybody’s guess. This, because nobody is are apt to play a machine enough to get close to the theoretical frequencies other than by coincidence. A game that seems hot to one solid citizen may therefore seem cold to the next person, or even to the same player at a different time.

The accompanying table, giving the data for an actual 92.61 percent return machine, can be used to illustrate why short-term observations mask the actual characteristics of the slots. Given the number of stops on each reel, this game has 373,248 ways to form final combinations. A player would have to undergo many multiples of 373,248 spins to obtain a distribution of results which began to home in on the theoretical proportions of the various payouts. For instance, in 373,248 spins, which some folks might consider a “cycle,” the most likely number of 833-for-1 jackpots hit is eight. At a sustained average feverish pace of one round every five seconds, this many spins represents about 130 four-hour sessions. Even here, however, the likelihood of hitting exactly eight jackpots is only 13.96 percent. The chance of no jackpots is low – 0.03 percent. Fewer or more than eight jackpots come in at 45.30 and 40.74 percent, respectively.

Probabilities and contribution to overall player return percentage for a representative real slot machine

             ways                         return		
return     to hit     probability     percentage	
0         319,928        85.7146%        0.0000%		
2          18,960         5.0797%       10.1595%		
5          24,354         6.5249%       32.6244%		
10          7,198         1.9285%       19.2848%		
20          1,510         0.4046%        8.0911%		
25            336         0.0900%        2.2505%		
40            274         0.0734%        2.9364%		
50            362         0.0970%        4.8493%		
80            110         0.0295%        2.3577%		
100           104         0.0279%        2.7864%		
160            80         0.0214%        3.4293%		
320            24         0.0064%        2.0576%		
833             8         0.0021%        1.7851%		
overall   373,248       100.0000%       92.6121%	

Pretend you play just one four-hour session, keeping your fingers flying and getting an average of one spin every five seconds. Your total number of rounds would be 2,880. In this span, the probability of not hitting a jackpot at all is 94.01 percent. That of hitting once is a small but still respectable 5.80 percent. Any number more than this is down to about 0.18 percent – not an impossible dream but not eggs to be counted before they hatch, either. The outlook for missing both the 833-for-1 jackpot and the 320-for-1 next highest return in 2,880 spins is 78.12 percent. Prospects of hitting one or the other once is respectable at 19.29 percent. For two of either or one of each, it’s 2.38 percent. More than this is down around 0.21 percent.

Note that 833-for-1 (2,500-for-3 with three coins bet) is generally regarded as a low jackpot. Many machines dangle far more dough at the end of the stick. You needn’t be a gambling guru to guess that the richer the reward, the tougher it is to win. The returns shown in the table could be changed to pay a 3,333-for-1 (10,000-for-3) jackpot, leaving everything else essentially the same, by providing two rather than eight winning combinations for jackpots and 319,934 rather than 319,928 ways to lose. The return percentages would be equal to within two decimal places on the alternate versions. At 3,333-for-1 with two out of 373,248 ways to win, the chance of missing a jackpot in 2,880 spins would increase to 98.47 percent. The probability of one jackpot in 2,880 spins would be 1.52 percent, and of two or more jackpots only 0.01 percent.

Another factor to consider is the effective return percentage if you don’t hit a jackpot. On both the 833-for-1 machine with 8 out of 373,248 or the 3,333-for-1 game with 2 out of 373,248 ways to win, the jackpot contributes 1.78 percent to the return. So, instead of a 92.61 percent return, you’d be playing at 92.61 - 1.78 = 90.83 percent game. Worse, yet, if you also never got a hit at the second-highest level.

Casino games other than slots have much smaller numbers of possible outcomes. Less action is therefore required to start approaching the theoretical distribution of results. Not that this is anything to be desired, because the theoretical distribution is what gives the casinos their edge. As the renown rhymster, Sumner A Ingmark, rhetorically asked:

Gamblers and bosses have goals antithetical,
Guess who prevails with results theoretical.

Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.