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# Can you make sound gambling decisions not knowing the odds of winning?

16 April 2012

Gamblers selecting games to play and bets to make should ideally consider both payoffs and probabilities. Either might be the dominant factor for individuals picking from among alternatives. One person might opt for scoring big despite adverse odds. Another might choose frequent wins regardless of small returns. Together, payoffs and probabilities paint a comprehensive picture. To envision the effect, say two \$5 wagers each paid \$50. Would you rather take a shot overcoming odds of 11-to-1 or 12-to-1? Likewise, pretend the probabilities on two alternatives were one out of six. Would you prefer to risk \$10 on a bet paying \$35 or \$40?

The dilemma isn’t always this clear cut. Make believe you’re planning to bet \$5 at craps. On a nine, it’ll win \$7 against 6-to-4 odds; on a 10, it’ll pay \$9 fighting 6-to-3 odds. Apples and oranges? The gurus combine payoffs and probabilities to yield house edge or advantage – or, for machines, player return percentage – as a way to compare disparate bets. Advantage and edge are the fraction of the money wagered over many coups that the casino keeps from the net of wins and losses. Return percentage is the complement, the fraction that bettors collectively get back.

In the most basic cases, find edge or advantage as the gain per dollar bet times the probability of winning, minus the loss per dollar bet times the probability of losing. For a nine at craps, the value is (7/5) x 4/(4+6) - (5/5) x 6/(4+6) = -0.040 or -4.0 percent; for a 10 it’s (9/5) x 3/(3+6) - (5/5) x 6/(3+6) = -0.067 or -6.7 percent. The minus signs show that the money goes to the house. As player return, the figures would be 100.0 - 4.0 = 96.0 percent on nine and 100.0 - 6.7 = 93.3 percent on 10. Obviously, the more the bosses pocket, the less favorable the game is for bettors.

Players usually know the payoffs for their winning bets. The amounts are often traditional and standardized – like 7-to-5 on the nine or 9-to-5 on the 10 at craps, 35-to-1 for a single spot at roulette, or 3-to-2 for a natural at blackjack. If not, they’re prominently displayed on the layouts or placards next to the dealer at the tables or on the faceplates or video displays at the machines.

For many games, the probability of winning is less up-front but no great secret. It sometimes depends on self-evident factors such as the number of grooves on a roulette wheel, the totals that can be formed from a pair of dice, or the population of cards in a deck or shoe. Chances may also hinge on rules imposed on dealers. Illustrations are the totals on which they must stand or hit at blackjack or draw third cards for Player and Banker at baccarat. More complex situations arise under circumstances when probabilities are influenced by strategies solid citizens elect to follow. Illustrations are decisions about what to dump or hold when drawing at video poker or whether to stand, hit, double down, or split a hand against a given dealer upcard at blackjack.

Slots present the exception. Probabilities associated with symbols on a reel can be assigned arbitrarily. And there’s no way to tell what they are, other than from design specifications – which are not normally made public. Even casinos don’t generally know the chances associated with returns on specific devices. Further, because the attributes of the games are programmed into computer chips, manufacturers can and do produce multiple versions of machines which appear identical but have distinct arrays of probabilities.

As an example, seven configurations of the five-reel video Lobstermania game are approved for casinos in Ontario, Canada. The permutations have player returns ranging from 85.0 to 96.2 percent. A line of symbols has the same payout on all models – for instance, 2-for-1 for lobsters on the first and second reels, through 1,000-for-1 for lobsters in all five positions. Numbers of lobster stops programmed on corresponding reels differ among versions, however. This affects the probability of the various results. The first through fifth reels have 4, 3, 3, 3, and 4 such stops on the 85.0 percent machine and 4, 4, 3, 4, and 4 on the 96.2 percent model. The chance of a lobster on any reel is the number of lobster stops divided by the totality of stops on the reel. The overall prospect at each win level is the product of the probabilities associated with the requisite number of contributing reels. The chance is always less on the lower-return game. For instance it’s 0.55504 and 0.74066 percent, respectively, to finish with lobsters on the first two reels, up to 0.00017 and 0.00030 percent to find the coveted little red decapods on all five reels.

Slot players are unaware of chances of various payouts. So they know neither the likelihood of winning on any spin, nor the overall return percentage of a game. They may therefore find a “lucky machine” one day. But will they be able to identify it on their next visit? As the poet, Sumner A Ingmark, lamented:

A gambler unwary may quickly go broke,
By picking a game like a pig in a poke.