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Playing It Smart: How to get an edge at Pluto Craps

11 March 2008

Resolving world crises at corner taprooms or coffeehouses can grow old. So, many solid citizens look for friendly gambles at their tables or the bar, preferably where they can have an edge when nobody is the wiser. Pluto Craps may be just the thing.

The game requires one or two dice and a shaking cup (to prevent dice setting). When two people play, one calls the point and the other shakes and rolls first. If the point hits, the roller wins. If not, the other person takes a shot. The cup passes back and forth until someone wins. They switch roles for the next coup.

Is there an edge? If so, how much and who has it? As in all gambling, the laws of probability hold the answers. Here's how to find them. Label the probability of winning on any roll as P and that of not winning on the roll as Q=1-P. If you roll one die, P is one out of six (1/6) and Q is the other five out of six (5/6).

To simplify the discussion, name the players alphabetically. Amy rolls first so Zeke calls. The probability Amy wins right off the bat is P; the chance she doesn't is Q. The likelihood Zeke gets to roll after Amy misses is Q; the chance he'll win then is P. Zeke's overall chance of getting a first roll and hitting is QxP.

Extend this reasoning. The chance Zeke won't win on his starting toss is Q. If he doesn't, there have been two no-win rolls in a row (the chance of which is QxQ), and Amy recovers the cup for a new turn. She then has a chance of winning equal to P. Similarly, Zeke will get a second try after three straight no-wins QxQxQ, then has probability P of winning. Continue and combine terms for an infinite series since neither may ever make the point.

This gives Amy's ultimate chance as P + QxQxP + QxQxQxQxP + ... each term being multiplied by Q twice more than the last. Analogously, Zeke's ultimate chance is QxP + QxQxQxP + QxQxQxQxQxP + ... The math mavens have a shortcut to do the arithmetic for these two series. You don't even need a calculator and can obtain the results if you remember how to do fractions and long division. Amy's chance is 1/(1+Q) and Zeke's is Q/(1+Q).

With the single-die version of the game, P = 1/6 and Q = 5/6. Amy's chance is (1)/(1+5/6) = 6/11 = 54.5 percent. Zeke's chance is (5/6)/¬(1+5/6) = 5/11 = 45.5 percent. Whoever rolls first has the advantage, 54.5-to-45.5. Divided out, that's 1.2-to-1.

Using two dice, prospects depend on the number called. For instance, the chance of a two or 12 is one out of 36 (1/36); that of a seven is six out of 36 (6/36) = (1/6). The first roller still has an edge but it's 50.7-to-49.3 (1.03-to-1) on a point of two or 12, and 54.5-to-45.5 (1.2-to-1) on a point of seven.

Although the first roller always has an edge, you alternate from coup to coup. But you can still gain an advantage or have no worse than a fair game. When you call the point, pick two or 12. Your opponent's edge is then only 1.03-to-1. If your opponent also calls 2 or 12, the game is balanced. If your opponent calls anything else, you have a greater advantage when you roll first giving you the edge on the average.

Pluto Craps can involve more than two players. Here, the previous first roller calls the point and is last at bat in the new coup. The first roller has the best chance to win, followed by the second, and so on. With two dice, the probabilities again depend on the selected point. Take four players as an example. For a seven (P=1/6, Q=5/6), chances for the first to the fourth roller work out to be 32.2, 26.8, 22.4, and 18.6 percent, respectively. For a two or 12 (P=1/36, Q=35/36), chances are 26.1, 25.3, 24.6, and 24.0 percent, respectively. When you're calling, choose two or 12 and hope the others pick more readily-rolled points.

You can have fun with Pluto Craps. And, even if you never play, it can help you understand that mathematics underlies the games you find in a casino. While anybody can be lucky, those who know the numbers typically make better choices than those who don't. It's as the awesome odester, Sumner A Ingmark, observed:

To do more with your gambling urge than just humor it,