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Why slots that look identical may differ greatly

31 October 2011

Everyone knows, or should know, the returns for bets they make in the casinos. On propositions that win, lose, or push, earnings are fixed multiples of the money at risk. On those with diverse possible outcomes, such as video poker hands or reel-type slot machine symbol alignments, returns follow predetermined schedules based on results obtained.

But payoffs aren’t the whole story. The associated probabilities are also important. Ask yourself whether you’d prefer an even-money wager with 49 percent chance of winning versus 51 percent of losing, or one with respective probabilities of 48 and 52 percent.

Payoffs and probabilities, together, are necessary and sufficient to fully characterize bets. They’re combined to yield the theoretical advantage or edge the bosses have over bettors – a figure many sophisticated solid citizens use in selecting what and how to play. They also yield attributes other than edge, that are less well recognized but as or more critical from a player’s perspective.

Reel-type slots offer a way to illustrate the relevance of probabilities along with payoffs, and how these factors can be used in concert to help bettors become happier campers. For this purpose, picture a group of three such machines, all showing the same payout schedule. You might assume, based on the payoffs, that the games are identical. This could but may not be the case. Edge (on the slots it’s usually expressed in a complementary form as return percentage), may differ among the machines. Moreover, irrespective of edge, probability distinctions may affect prospects of winning a jackpot, hit rate – the frequency at which any returns can be expected over the course of time, and prospects of survival – the chance that a specified bankroll will suffice to give players good runs for their money.

A virtually infinite number of probability combinations is possible for these devices and there’s no a priori way deduce what they are. The disparities can occur even if the machines have the same house advantage. As an example, the configurations in the accompanying table are unique, yet all have the same payout schedules and equal 2.5 percent edges – 97.5 percent player returns.

Payouts and alternative probabilities for three visually identical 97.5 percent slot machines
     Aligned                  Probabilities                   
     symbols      Return  Game 1  Game 2  Game 3
        eagles      1000  0.001%  0.001%  0.001%
     cardinals      100   0.002%  0.014%  0.129%
        robins      50    0.004%  0.028%  0.257%
         hawks      25    0.469%  0.055%  0.514%
         crows      10    0.938%  0.111%  1.028%
       pigeons      5    10.000% 10.750%  2.055%
      sparrows      1    25.000% 37.500% 37.500%

As far as the casino is concerned, these devices are indeed equivalent. Over enough trials, hundreds of thousands or millions, the law of averages prevails and the casino will net close to the edge, 2.5 percent, multiplied by the gross amount wagered.

No particular solid citizen will get anywhere near the amount of action needed for edge to be the determining factor in their play. Hit rate and probability of survival rise to the fore during statistically short sessions, casino visits, or even series of relatively frequent gambling trips. The hit rate is especially evident to slot buffs. Most such players find higher hit rates encouraging and lower frequencies of returns discouraging, independently of the money involved. It typically doesn’t take long for people experiencing one or the other end of this spectrum to conclude that a machine is hot and worth continuing effort or cold and best abandoned, notwithstanding the actual ebbs and flows of their bankrolls. On game 1, the hit rate is 36.4%, approximately one out of every 2.75 spins. On game 2, it’s 48.5 percent, about one out of every 2.1 spins. On game 3, the hit rate is 41.5 percent, roughly one out of every 2.4 spins.

As for likelihood of survival, picture hopeful gamblers with $100 bankrolls who bet $1 per spin on the prototype machines. Say these folks get 10 spins per minute and – even if they lose – think they got their money’s worth of entertainment as long as they don’t go bust before experiencing four hours of excitement. On game 1, the chance of surviving the 2,400 spins is 30 percent. It’s somewhat better on game 2, at 31 percent. And it’s worse on game 3, at 23 percent.

Of course, games that look identical would differ even more were they not to all have the same return percentage. In particular, higher edge lowers players’ prospects of surviving action of any desired period with stakes they considered sensible before leaving home.

So, what you think you see in the casino may not be what you’re gonna get. Or, as the punter’s poet, Sumner A Ingmark, prophetically penned:

In atmospheres of smoke and mirrors,
Apparent truths are fraught with errors.
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.