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Best of Alan Krigman

Gaming Guru

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Playing It Smart: When it comes to slots, what you see isn't what you get

27 May 2008

Picture a slot machine with three reels, each having 10 symbols and 10 corresponding points at which to stop. If each symbol was unique, the machine could display 10 x 10 x 10 or 1,000 combinations. The probability of any one combination would be (1/10) x (1/10) x (1/10) or 1/1,000. This equals 0.1 percent.

Experience shows that actual games have fewer combinations, with varying likelihoods of occurrence. This is achieved in two ways.

First, the reels may have one or more instances of some symbols. Say a tiger did occur only once per reel. With three reels and 10 positions per reel, the chance of three tigers would be 0.1 percent. If this paid 50-for-1, the contribution to the return rate would be 0.1 percent of 50, which equals 0.05 or 5 percent. But this device might also have three dogs and one pony on the first and second reel and four dogs and two ponies on the third.

Perhaps you'd guess that the chance of two dogs and a pony in any order is 4.2 percent based on the following math. Dog-dog-pony: (3/10) x (3/10) x (2/10) or 18/1,000; dog-pony-dog: (3/10) x (1/10) x (4/10) or 12/1,000; or pony-dog-dog: (1/10) x (3/10) x (4/10) or 12/1,000. Add 'em up to get 42/1,000 – 0.042 or 4.2 percent. Extending this reasoning, if this result paid 3-for-1, it would contribute 3 x 4.2 or 12.6 percent to the overall payback percentage of the machine. Maybe. Or maybe not.

Maybe not, because the slots, even those with what look like simple mechanical reels, don't necessarily have one stop per symbol. Symbols may have multiple stops, with mechanisms to center them in the windows regardless of the actual points at which the reels catch. And, numbers of stops can differ among symbols. This lets casinos adjust the probabilities while keeping solid citizens from deducing the statistics of the game. Counting symbols doesn't help without knowing numbers of stopping points.

Picture a reel with 10 symbols and 100 stopping points. Players might see one tiger on each reel. The tiger might have only one catch, although players would have no way to tell. If it did, the chance of three tigers would be (1/100) x (1/100) x (1/100) or 1/1,000,000. This result could pay 50,000-for-1 and contribute the same to the overall payback rate as 50-for-1 does on the machine with 10 stopping points. That is, 50 x (1/1,000) and 50,000 x (1/1,000,000) both equal 0.05 or 5 percent. Similarly, edge wouldn't change if the tiger on one reel had two catches, doubling the probability of a hit, but the payoff was halved.

Pretend also that on each reel, dogs had 10 stops and ponies had five each. Extending the previous example, the chance of two dogs and a pony would be dog-dog-pony: (10x3/100) x (10x3/100) x (5x2/100)or 9,000/1,000,000; dog-pony-dog: (10x3/100) x (5x1/100) x (10x4/100)or 6,000/1,000,000; or pony-dog-dog: (5x1/100) x (10x3/100) x (10x4/100) or 6,000/1,000,000. That adds up to 21,000/1,000,000, or 2.1 percent. At 3-for-1 this would contribute 3 x 0.021 or 6.3 percent to the payback rate.

Video slots carry this principle further. The reels are virtual and can have huge numbers of symbols and stopping positions. To get the idea, imagine a screen laid out like a tic-tac-toe game – three rows and three columns. If an X or an O could appear in any spot, that's only two symbols, there are 512 different arrangements. Were it possible to have X, O, or a blank – essentially three symbols, there are 19,683 arrangements. A multi-line multi-coin game with three rows and five columns, having only three symbols, could be configured in 14,348,907 different ways. And, of course, there may be 10 or a dozen symbols – before you even start worrying about stopping points and probabilities of each.

Little wonder most players prefer to follow this guidance of the punters' poet, Sumner A Ingmark:


When numbers get too big, conclusions may be doozies,

To keep your logic straight, envision onesies-twosies.
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.