Why Being Too Clever by a Half Doubles the Casino's Edge

Every so often -- too often, actually -- craps players "discover" how to beat the casinos by betting both Pass and Don't Pass then taking or laying odds on one or the other. Their theory is that the flat bets are self-canceling, except for the rare loss on the come-out when a 12 appears. And the house gets no juice on the odds portion of the bet. So, ignoring that occasional 12, players believe they get to make bets with no house advantage. Further, those who bet both sides and lay odds on the Don'ts can chortle securely in the knowledge that the seven dreaded by mere mortals can't hurt them, pushing on the come-out and winning thereafter.

Solid citizens don't always discover this loophole in the laws of mathematics on their own. They sometimes see it in a book so they know it's gotta be true. And, some casinos reportedly don't allow simultaneous Do and Don't bets, which fact -- or rumor, anyway -- strengthens the sense of a secret system for select insiders.

In reality this ill-conceived gimmick, far from eliminating the house advantage on craps line bets, exacerbates it. Betting both sides is the worst of both worlds. To picture the situation intuitively, you can consider only the flat portion of the wager because the odds don't affect the amount of the casino's theoretical "take." Normally, Pass bets have an advantage before the point is established, since they are paid even-money but have eight ways to win and only four to lose. Conversely, Don't Pass bets have an advantage after the point is established, since they pay even money but have six ways to win on any number with only three to lose on the four or 10, four on the five or nine, and five on the six or eight. Equal amounts on Pass and Don't Pass offer no way to win before or after the point is established, and one way to lose whatever's on the Pass line during the come-out.

The folly of betting both sides can also be expressed in numbers -- percentages as well as dollars and cents. The house edge is 1.41 percent of the flat bet on Pass and 1.36 percent on Don't Pass. That's roughly $0.07 for $5 bet on either side. With $5 on both sides, the take is $0.07 twice, for a total of $0.14.

To see exactly how edge works, imagine 1980 statistically-correct come-out episodes. Instances of wins and losses for Pass and Don't Pass bets are as shown in the accompanying table.

Wins and Losses in 1980 Statistically-Correct Come-Out Episodes
	Bets on Pass		Bets on Don't Pass
	win	lose	win	lose
Line	440	220	165	440
6/8	250	300	300	250
5/9	176	264	264	176
4/10	110	220	220	110
total	976	1004	949	976

For bets on Pass, expectation is to lose a net of 1004 - 976 or 28 units during the 1980 episodes. At $5 each, this amounts to $140. And 28 divided by 1980 is 1.41 percent, by no coincidence the house edge on Pass bets. Likewise for Don't Pass, expectation is to lose a net of 976 - 949 or 27 units during the 1980 episodes. At $5 each, this would be $135. And 27 divided by 1980 is 1.36 percent, also by no coincidence, the edge on Don't Pass.

A player who bets $5 each on Pass and Don't Pass will lose $5 in those rare cases when 12 rolls on the come-out. How rare? On the average, once every 36 times. Divide 1980 by 36 to get 55 as the number of 12s in 1980 statistically-correct come-outs. And $5 times 55 is $275, which happens to be the sum of the expected $140 and $135 losses on the separate $5 Pass and Don't Pass bets.

What about the odds? This portion of the bet has no impact on the amount of expected loss. Rather, it increases the money players can put in action for the fee the casino hides in the form of the edge. The whole idea in a negative expectation game is to get the most action for the least fee, depending on volatility for a profitable session. Betting Pass and Don't Pass together, then taking or laying the odds on either, defeats this objective.

Here's how the poet of the pits, Sumner A Ingmark, put it.

Bettors who act too clever by a half,
Give the casino bosses the last laugh.