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# Which numbers at craps are better to Buy than Place?

16 August 2010

Some craps buffs wager directly on fours, fives, sixes, eights, nines, or 10s. They win if the dice show the selected values and lose on sevens. The makes odds against winning 6-to-3 (2-to-1) on fours or 10s, 6-to-4 (3-to-2) on fives or nines, and 6-to-5 on sixes or eights. The bets have two modes: Placing and Buying.

On fours or 10s, Place bets are made in \$5 increments and pay 9-to-5. On fives or nines, the bets are also \$5 multiples and pay 7-to-5. On sixes or eights, they're in \$6 steps and pay 7-to-6.

Buy bets pay the precise inverse to the odds of winning -- with a catch. The house charges a "vigorish" or "vig," nominally 5 percent of the wager. The fee is usually levied up front and the house keeps it, win or lose. Effectively, solid citizens put up the bet plus the vig to net the payoff on the bet minus the vig.

Owing to standard craps practice that bets and payoffs are in whole dollars, casinos round the vig downward by dropping the cents. Savvy players can exploit the fact that the actual vig falls below 5 percent for bets that aren't integral multiples of \$20. A true 5 percent commission would cost players \$1 on \$20, \$2 on \$40, but \$1.95 on \$39. If you Buy the four for \$39 rather than \$40, the casino rounds the \$1.95 down to \$1, which is about 2.6 rather than 5 percent of the wager. As another implication of the round-off rule, \$20 is the minimum for Buy bets because 5 percent of anything less rounds down to zero.

The "whole dollar" rule also governs the increments in which bets can be made. Fours and 10s pay 2-to-1; so any whole-dollar amount is allowed since 2-to-1 payoffs will also be in whole dollars. Fives and nines pay 3-to-2; therefore, only bets in even dollar amounts can be paid in whole dollars, so this constraint is in force. Sixes and eights pay 6-to-5, so bets must be in \$5 bites.

If you're gonna bet \$20 or more on a number, should you Buy or Place it? The chance of winning is the same. As a result, the better option is that which gives the house less in edge. Picture this in terms of 36 statistically-correct rolls of the dice.

Placing the four or 10 for \$20 would lose \$20 six times and win \$36 three times -- giving the house \$120 - \$108 = \$12. A \$20 Buy would put up \$21 to net \$39 -- giving the house \$126 - \$117 = \$9. At this level, the Buy is superior. Buys for larger bets either decrease the house's relative take further owing to the rounding-down, or leave it at appropriate multiples of \$9.

Placing the five or nine for \$20 would lose \$20 six times and win \$28 four times, giving the house \$120 - \$112 = \$8. A \$20 Buy would put up \$21 to net \$29 -- giving the house \$126 - \$116 = \$10. At \$20, Place bets are accordingly preferable. A \$25 Place bet would give the house \$150 - \$140 = \$10, while a \$24 Buy would put up \$25 to net \$35 -- also giving the house \$150 - \$140 = \$10. Buys of \$26 through \$38 would earn the house proportionately less and consequently be superior to their Place counterparts. There are also bets over \$40 where Buying beats Placing.

Placing the six or eight for \$24, the lowest multiple of \$6 that could qualify for a Buy, would lose \$24 six times and win \$28 five times -- giving the house \$144 - \$140 = \$4; Buying the same number for \$25 would mean putting up \$26 to net \$29, giving the house \$156 - \$145 = \$11. The \$24 Place bet would be far superior. The best Buy bet on six or eight would be \$35, putting up \$36 to net \$41. The house would get \$216 - \$205 = \$11 from this situation. Placing the number for \$36 to win \$42 would give the house \$216 - \$210 = \$6, so the Place bet is still preferable.

When the vig is paid up front, Buy bets always give the house less edge on fours and 10s than similar Place bets. On sixes and eights, Place bets always yield less edge. On fives and nines, some bet sizes do better Buying and others Placing. So, Buy fours and 10s, Place sixes and eights, and think before betting on fives and nines. As the inkster, Sumner A Ingmark, intoned:
If winning's what your heart is set on,
Be sure you know what's best to bet on.