What Casinos Can't Predict about Jackpots, and What They Can

The only thing known about a jackpot, other than amount and increment if it's progressive, is its chance of being won on any try. This, because an internal computer chip generates numbers at random and with equal likelihood over a predetermined range, a specified set of which corresponds to the jackpot. Say the chip spits out values from 1 to 1,000,000, with numbers 256 and 32,768 triggering the biggie. Prospects of winning would be two out of a million or one out of 500,000, a 0.0002 percent probability.

The chance of a jackpot doesn't tell when it will hit. But it gives casinos other data they can use in running their operations effectively. In particular, it lets them know how often they should expect jackpots, over the long run, and the likelihood of various numbers of occurrences during any designated time spans.

As an example, pretend that your favorite casino is thinking of installing a bank of 30 Little Bo Peep slots. The bosses will be thrilled if the units come home with six hours of action a day at 250 spins an hour. This would be 1,500 spins per machine per day, times 30 machines, times 30 days or 1,350,000 spins per month.

Maybe the manufacturer offers standard models having jackpots of $12,500 with probabilities set at four out of 1,000,000, $25,000 at two out of 1,000,000, or $50,000 at one out of 1,000,000. Halving the chance and doubling the amount keeps the edge constant. In this sense, the casinos view the options as a wash.

The slot honchos can decide whether the games will be more apt to draw at least the desired action if they have smaller jackpots but gain a reputation for frequent payouts, or the opposite, or somewhere in between. They can also consider two other factors: 1) the risk of hurting the company's short-term profits by featuring high jackpots but ending up paying them more often than forecast by the long run averages; 2) the danger of opting for low jackpots then paying them so rarely that players get turned off and it seems as though Little Bo Peep has lost her sheep and doesn't know where to find them.

The primary calculation is how many jackpots are expected to be paid per month. This is easy: multiply the probability times the contemplated number of spins to get 5.4 jackpots at $12,500, 2.7 at $25,000, and 1.35 at $50,000. The other two factors take some fancier statistical footwork. The math mavens could use either a binomial or Poisson distribution, depending on how they frame the question (results are equivalent in this case), to ascertain the chances of up to eight jackpots per month at each of the three payout levels. The figures are shown in the accompanying chart.

Probability of from zero through eight
jackpots per month at the
$12,500, $25,000, and $50,000 levels
for the indicated machine configurations

The most likely numbers of hits bracket the expected values. For instance chances are 24.5 percent for two and 22.05 percent for three at $25,000. But a casino shouldn't start looking for a conspiracy if it pays three or more $50,000 jackpots in any given month when the average is 1.35, since the combined probability is at this level is a respectable 15.46 percent. Nor should it be too surprised if nobody hits for $25,000 in the same time span because the chance here is a small but real enough 6.72 percent. As the reckoning rhyme writer, Sumner A Ingmark, recounted:

Projections based on probability,
When tossed around without agility,
Are exercises in futility.