Proposition Bets at Craps Are Appealing Despite the High Edge

"Propositions" or "props," listed in the accompanying table, are one-roll craps bets. They win or lose every time the dice are called. Casinos all offer prop bets on the two, three, seven, 11, and 12. In some jurisdictions, casinos also book propositions on "hops" -- bets that the next roll will show a specific pair of numbers such as three and five, one and four, or two and four.

One-roll proposition bets
Result	odds against winning	payoff	edge

two	35-to-1	30-to-1	13.9%
three	17-to-1	15-to-1	11.1%
seven	5-to-1	4-to-1	16.7%
11	17-to-1	15-to-1	11.1%
12	35-to-1	30-to-1	13.9%
hop	17-to-1	15-to-1	11.1%

Propositions can be bet singly, for instance $1 on the three or $1 each on the two and three. "Any craps" is a consolidated prop bet. This can be made for as little as $1 and pays 7-to-1 if the roll is a two, three, or 12; odds against winning are 8-to-1, payoff is 7-to-1, and edge is 11.1 percent.

A second group of consolidated prop bets is combined in name only. That is, you make the wager with a single instruction to the stickman. However, you're actually betting on the components separately and get paid according to the particular result of the roll. The primary combinations are as follows, but variations on the theme are sometimes requested (generally by dice devotees who have sent away for one secret system, read one article or book, have seen one mobster movie, or have had one daydream too many).

High-Low:	This bet, an even amount, is spread equally on the two and 12. A win gets the usual 30-to-1 on whichever number hits, but the complement loses. So, say you bet $2. The chance is two out of 36 you'll net $29, versus 34 out of 36 you'll lose $2, giving the house a 13.9 percent edge.
Horn:	This bet must be an exact multiple of $4 -- $4, $8, $12, 16, 20, and up. A fourth of the total goes on each of the two, three, 11, and 12. For every $4 you risk, the chance is two out of 36 that the roll will turn up two or 12 and net you $27, four out of 36 that the dice will show three or 11 and net you $12, and 30 out of 36 that another number will appear and cost you $4, giving house a 12.5 percent edge.
Horn High ###:	The horn can also be bet, not in multiples of $4 but of $5 -- $5, $10, $15, $20, $25, and up -- adding the increment to the selected spot. For instance, "a nickel horn high aces" is $1 each on the three, 11, and 12 with $2 on two ("nickel" being wiseguy-ese for $5 and "aces" for the two). Similarly, "a quarter horn high yo" would be $5 each on the two, three, and 12 with $10 on 11 ("quarter" and "yo" being how righteous readers of Damon Runyan would refer to $25 and 11, respectively). Chances of winning are the same as for the uniform horn, but payoffs change according to the placement of the extra money. Edge is 12.8 percent on horn high two or 12 and 12.2 percent on horn high three or 11.
World (sometimes called whirl):	The props can be bet in multiples of $5, equally split on the two, three, seven, 11, and 12. For every $5 so wagered, chances are two out of 36 to net $26 on the two or 12, four out of 36 to net $11 on the three or 11, six out of 36 to push with a seven, and 24 out of 36 to lose everything. This yields an edge of 16 percent.
C & E:	This bet, which need not be an even amount, is split with half each on "any craps" and "yo." For every $1, the chance is four out of 36 to net $3 on a craps, two out of 36 to net $7 on an 11, and 30 out of 36 to lose. Edge is 11.1 percent.

House advantage is high on the props. Yet, they're popular. Why?

The theory of utility explains that rational folks will flout the math and take risks if they perceive the reward to be high and the penalty low. The miasma of misanthropy adds that flinging money across a table and shouting out a command is an exercise in assertiveness for solid citizens who resent routine repression by spouses, bosses, in-laws, offspring, gambling gurus, and others of the power elite. Take your pick. The poet, Sumner A Ingmark, no stranger to skepticism, supported the more cynical stance:

Freed from jeering, freed from sneering,
Milquetoasts become domineering,
Hooting, hollering, and cheering.