Craps Players Can Tailor Sessions without Changing Expectations

8 February 2006

By Alan Krigman

Make believe you sent away for a book of secrets the casino bosses didn't want anyone to know about craps. Or maybe it was secrets the bean counters and back slappers in the head office didn't understand enough about craps to know themselves.

The book told how to cut the edge on Place bets. The premise was that betting either the five or nine would give the casino 4 percent edge. But betting equal amounts on both the five and nine would shave the house advantage to 2.8 percent. And similar reductions could be achieved with other Place bet combinations.

The claim is true but irrelevant. Casinos calculate edge on these wagers idiosyncratically. The value is based on number of actual decisions and ignores throws of the dice that don't win or lose.

Here's how it works. Say the dice roll 36 times. Think of each toss as a separate bet. The "statistically correct" distribution of results includes six sevens, four fives, and four nines.

Bet $10 on five. You'd have four $14 wins and six $10 losses, 10 total decisions. You'd pick up $14 x 4 = $56 and drop $10 x 6 = $60 for a net loss of $4. Since there were 10 decisions, the casino figures you had $10 at risk for 10 coups, a handle of $10 x 10 = $100. The edge is $4/$100 or 4 percent.

Instead, bet $5 each on five and nine. You'd have eight $7 wins and six $10 losses, 14 total decisions. You'd pick up $7 x 8 = $56 and drop $10 x 6 = $60 for the same $4 net loss. But, with 14 decisions, the casino figures you had $10 at risk for 14 coups, a handle of $10 x 14 = $140. Edge is then $4/$140 or 2.8 percent.

The lower percentage associated with $5 each on five and nine rather than $10 on one or the other doesn't mean you're less of an underdog. Although the numerical values of edge differ, so do the handles by which they're multiplied. And 4 percent of $100 equals 2.8 percent of $140, giving the casino the same $4 theoretical "expectation" either way. And expectation in dollars, not edge in percent, is what depletes your fanny pack.

Simulation of 3,600,000 throws verifies that these values are reliable averages. But, at 100 throws per hour in a fast game, this represents 36,000 hours of action. No solid citizens play this much in their lifetimes. And they wouldn't play at all unless they thought they'd get lucky and beat the averages.

Inquiring minds will therefore want to know the effect of factors other than expectation which may follow from concentrating or spreading a given total bet around a craps layout. Especially, whether they influence the chances associated with sessions or visits of reasonable duration.

A simulation of 1,000 players, betting exclusively either $10 on five or $5 each on five and nine for 360 throws, about four hours, offers some insights. The theoretical average across all players is a loss of $40 one way or the other. The simulation yielded average losses of $40 for five-only and $41 for five and nine, not appreciably distinct for a 1,000-player sample. Of those with $10 on the five, 317 finished profitably, 675 ended in the hole, and eight broke even. Bettors who put $5 each on five and nine had 282 triumphs, 712 defeats, and six break-evens. Medians, points at which half did better and half worse, were losses of $43 and $45 for the one- and two-bet strategies, respectively. The maximum profits were $278 with money on just the five and $218 on the five and nine while the worst deficits were $306 for five-only and $264 for the combination bet.

Short-term performance differed, despite equal expectations. Looking for lower-probability bigger hits resulted in greater session earnings but also larger losses than the alternative. It also yielded more winners and fewer losers. Suggesting that edge is only one of many rabbits in a gambler's hat. Or, as the metrical muse, Sumner A Ingmark, memorably murmured: