How much do Buys at craps improve, paying the vig only on wins?

Buy bets on four through six and eight through 10 at craps are of two types. Most commonly by far, the house collects a "vigorish" or "vig" up-front, when players make their wagers. A few casinos, regularly or as special promotions, take the fee out of the payoff only if and when the wager wins. The nominal vig either way is usually 5 percent of the bet rounded down to the next lower whole dollar. And there's a $20 minimum because 5 percent of anything less, rounded down to an integral value, is zero.

Buy bets, apart from the vig, pay in exact proportion to the odds against winning. Without a margin between the odds and the payoff, the house needs the vig to gain an advantage or edge.

Here's what happens on up-front vigs. If the wager loses, players forfeit the bet plus the vig. If the wager wins, the house pays and returns the bet but not the vig; players therefore earn the payoff minus the vig.

The vig-on-win situation is as follows. If the wager loses, solid citizens get hit only for the amount bet. If the wager wins, the house returns the bet and makes the payoff while withholding the vig; the net gain is accordingly the payoff minus the vig.

To compare the options, consider the totals players would lose betting $20 in 36 statistically-correct rolls of the dice.

Four or 10: With vig-up-front, players earn $39 ($40 - $1) three times and forego $21 ($20 + $1) six times; net loss is 6 x $21 - 3 x $39 or $9. Effective edge is the $9 loss divided by $21 put up in nine cycles, $9/(9 x $21) or 4.76 percent. With vig-on-win, players earn $39 three times and give up $20 six times; net loss is 6 x $20 - 3 x $39 or $3. Effective edge is the $3 loss divided by $20 put up in nine cycles, $3/(9 x $20) or 1.67 percent.

Five or nine: With vig-up-front, players win $29 ($30 - $1) four times and drop $21 six times; net loss is 6 x $21 - 4 x $29 or $10. Effective edge is the $10 loss divided by the $21 put up in 10 cycles, $10/(10 x $21) or 4.76 percent. With vig-on-win, players earn $29 four times and give up $20 six times; net loss is 6 x $20 - 4 x $29 or $4. Effective edge is the $4 loss divided by $20 put up in 10 cycles, $4/(10 x $20) or 2.00 percent.

Six or eight: With vig-up-front, players gain $23 ($24 - $1) five times and kiss $21 goodbye six times; loss is 6 x $21 - 5 x $23 or $11. Effective edge is the $11 loss divided by $21 put up in 11 cycles, $11/(11 x $21) or 4.76 percent. With vig-on-win, players grab $23 five times and give up $20 six times; loss is 6 x $20 - 5 x $23 or $5. Effective edge is the $5 loss divided by the $20 put up in 11 cycles, $5/(11 x $20) or 2.27 percent.

Effective edge is greatest when bets are integral multiples of $20 because the house takes exactly 5 percent. Between multiples of $20, rounding-down cuts the actual fraction of the bet taken.

The highest effective edge on a vig-on-win Buy for fours or 10s is 1.67 percent. This is much less than the 6.67 percent edge when these numbers are Placed, so all such Buys are better for bets of $20 or more. On fives or nines, the biggest effective edge on vig-on-win Buy bets is 2.00 percent -- half that of Place bets; all such Buys are again preferable for bets of $20 or more. On sixes and eights, the greatest edge on vig-on-win Buys is 2.27 percent, when the bet is $20. Place bets on six or eight have 1.52 percent edge, so the latter are the superior choice at this level. Effective edge on vig-on-win Buys at $25 is 1.82 percent, also more than on Place bets. Edge is 1.52 percent on both at $30. Edge gets less when Buying at $35, 1.30 percent. Over $40, rounding down reduces the effective edge on vig-on-win Buys but never below the 1.52 percent for Placing the number.

Craps buffs who want to minimize edge make Come, not Buy or Place bets. Come bets give the house 1.4 percent edge flat, less backed up with Odds. As for rejecting Comes because you can't pick your own numbers, or must hit a value twice before these bets win on points, the Dryden of the dice, Sumner A Ingmark, declared:
Craps decisions based on myth,
Lead to bankruptcy forthwith.