Even-Money Equivalents Show the Effect of Odds Multiples at Craps

4 October 2004

By Alan Krigman

Most craps aficionados know that casinos give them a better deal by offering higher Odds multiples on Pass, Come, Don't Pass, and Don't Come bets. However, surprisingly few dice devotees actually understand how this affects their chances, what's so great about it, and whether they should use the opportunities afforded.

So, people patronize a casino because of the Odds, then bet Place, Buy, and Lay rather than Come and Don't Come. Or bet less Odds than allowed. Some even go so far as to apportion their total outlays in a manner that diverts money from the Odds to put more than the minimum on the flat portions of their wagers.

The converse happens, too. Solid citizens may take or lay all the odds they can get, overbetting their bankrolls by doing so. These folks often have the pleasure of going home broke, secure in the knowledge that the bosses didn't make any money on their action.

Ultimately, the benefit of high Odds multiples is low house advantage. The casino has 1.4 percent edge on the flat parts of the associated bets. It has no edge on the Odds added after a point is established. So $10 flat gives the joint an expected profit of $0.14 (1.4 percent of $10). And $10 flat with $20, $50, $100, or $1,000 in Odds still has a $0.14 negative expectation -- in contrast to two, six, 11, or 101 times as much were the total directly on the line. Therefore, $10 with $20 odds trims the casino's edge to 0.61 percent. This is $0.14 divided by the sum of $10 plus two-thirds of $20 (two-thirds representing the 24 out of 36 possible come-out rolls when a shooter establishes a point. Likewise, for $50 (5X), $100 (10X), and $1,000 (100X) in Odds, the edge drops to 0.33, 0.18, and 0.02 percent, respectively.

But edge tends to be elusive in that the dealers don't explicitly deduct the corresponding amount from what's bet or paid. It's buried in the math of probabilities and payoffs. It's further obfuscated because, in sessions of reasonable duration, edge tends to bear less on gamblers' fortunes than does the volatility -- the inherent bankroll swings -- characterizing every bet.

One way to clarify the role of the Odds is to compare bets using their "even-money equivalents." These are hypothetical 1-to-1 wagers having sizes and probabilities of winning that yield identical edges and volatilities as their real counterparts. Even-money equivalents per dollar flat Pass or Come with various Odds multiples are given in the accompanying table.

Even-Money Equivalents Per Dollar Bet Flat
on Pass or Come with Various Odds Multiples

	0	2X	5X	10X	100X
Bet size	$1.00	$2.86	$5.82	$10.81	$100.79
Prob of winning	49.293%	49.894%	49.972%	49.991%	49.999%

Here's an illustration of how to interpret the table. Say you bet $10 on Pass and take $50 Odds. This costs or pays $10 during the come-out roll; subsequently, it costs $60 but pays $110, $85, or $70 and has different probabilities of winning, depending on the point. This proposition has the same edge and volatility as would a straight even-money bet of $58.20 ($5.82 x 10) with a 49.972 percent chance of success. According to this reasoning, betting $60 flat gives you a 49.293 percent chance to win $60; betting $10 and $50 odds is like betting $60 to win $60 with a much greater 49.972 percent chance of victory.

Inquiring minds, of course, will also want to know the even-money equivalents of the Place bets, per dollar wagered. The values are: four or 10, $1.32 with 47.478 percent chance of winning; five or nine, $1.18 with 48.300 percent chance of winning; six or eight, $1.08 with 49.298 percent chance of winning.

This approach to evaluating alternate wagers reduces what seem like disparate situations to a common ground. As such, it provides an intuitive picture of how bankroll fluctuations and likelihood of winning change from one bet to another. The former through the size and the latter the probability components. It's as the remarkable rhymester, Sumner A Ingmark, has written.

Comparing by equivalents,
Helps circumvent ambivalence.