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Best of Alan Krigman
What's Wrong with Odds-Only Hedges in Craps?31 May 1999
The other day, two craps players asked me to resolve a difference of opinion. Both these solid citizens had put on their thinking caps (probably the type with Cubs or Mets logos and visors turned backwards). They'd devised schemes to beat the casinos at their own games. Each planned to drop equal amounts on Pass and Don't Pass, then make a mint betting the "odds" on only one side.
The logic, they said, was so simple it was amazing nobody ever thought of it before. The flat wagers, on which the house had an edge, would more or less cancel each other out. As a result, they'd only be risking the odds. And the odds wager, they hastened to add, "has no edge for the house so it's one of the best bets in the casino." The dispute between them involved which way to go with the odds. One argued for taking the odds on the Pass line. The other advocated laying the odds on the Don't.
First, I explained that the premise behind their brainstorms was faulty. They weren't eliminating the house advantage at all. In fact, they were doubling it. Betting $10 each on Pass and Don't Pass, then going with the odds on one or the other, gave the casino the same edge as $20 on either side with half the maximum rather than full odds.
"How can this be?" they both asked. With their systems, the flat parts of the bet protect each other except in the few instances when 12 appears on the come-out; then, they lose $10 on Pass without getting it back on Don't Pass. On any other result, the bets are equal and opposite so they're a wash. "Correct," I countered. But one way in 36 to lose $10 on the come-out, with 35 ways in 36 to push and no way to win a cent, gives the casino the same theoretical profit as $20 on Pass or Don't Pass by itself. It works out to $0.28 for every decision cycle.
"Even if you're right," they responded, "which side is better for the odds?" I answered that the choices were virtually equivalent.
Edge is the usual way to compare wagers. And edge, as they had duly noted, is zero on either odds bet. The choice boils down to whether they'd rather win more money less often, or the converse.
As an example, say flat bets are $10, double odds are allowed, and the point is five. A decision is reached on any of 10 results -- the four ways to form a five and the six ways to total seven.
The bets are also equivalent from the viewpoint of characteristic bankroll fluctuations. These are described statistically in terms of "standard deviation" -- pictured as average bankroll change for each decision. Again, assume $10 bets with 2X odds. Standard deviations for the odds on either side are $21.90 for points of six or eight, $24.50 for five or nine, and $28.30 for four or 10.
There's a minor difference in what the math mavens call the "skewness" of the results. Long-term averages -- edge and bankroll fluctuations -- are the same. However, taking odds on Pass has a slight positive skewness; it tends to yield occasional big wins at the expense of more frequent small losses. Laying odds on Don't Pass has a low negative skew; it tends to yield greater numbers of modest wins, offset by sporadic large losses.
Bets on both sides of a proposition are "hedges." Craps offers numerous opportunities for them. They reduce a player's exposure on any roll, but increase the house advantage in the game. If hedging appeals to you, go ahead and do it. At last report, this was still pretty much a free country. But, don't delude yourself into thinking you've discovered some arcane secret the casino bosses don't want anyone to know. As the keen composer of craps couplets, Sumner A Ingmark, cleverly contended:
In those of math, you'll find no flaws.
Best of Alan Krigman