What's the edge on Buy bets at craps?

Casinos mint money by paying proportionately less to winners win than they collect from losers. The usual mechanism is the “edge” – a function of what few players realize is a ubiquitous offset between payoff ratio and odds against winning. As an example, at craps, a Place bet on the four loses six ways (a seven pops: 1-6, 2-5, 3-4, 4-3, 5-2, 6-1) and wins three (a four shows: 1-3, 2-2, 3-1). That’s 6-to-3 or 2-to-1 but the payoff is only 1.8-to-1. To envision how this generates real dough, say that over time, a casino books 90,000 $10 Place bets on the four. Players should win roughly 30,000 of these at $18 each for a total of $540,000, and lose the other 60,000 at $10 each for a total of $600,000. The house expects to net about $600,000 - $540,000 or $60,000.

The $60,000 net is 6.67 percent of the $900,000 gross wager. The 6.67 percent is the edge or house advantage on this particular bet. A different gross wager on the four would also yield close to 6.67 percent profit for the casino, provided only that it encompassed enough cycles for the law of large numbers to pertain. The actual formula for calculating edge is the probability of winning times the payoff, minus the probability of losing times the bet, divided by the bet. For the $10 four, this would be [(3/9) x 18 - (6/9) x 20]/20, which equals -6.67 percent.

The same principle, albeit with different figures, holds for other propositions. For instance edge is 2.7 percent on all wagers at single-zero roulette, about 0.5 percent at blackjack with decent rules when players follow Basic Strategy, and could be almost anything at the slots where the odds of returns at various levels are generally secrets the casino bosses don’t want you to know.
There’s a more overt way casinos come out ahead after long series of bets, some won and others lost by solid citizens, although it ultimately boils down to the same thing. Craps players willing to risk $20 or more on the four may, and ought to, Buy as opposed to Place the number. The odds fought are still 2-to-1. But the payoff isn’t 1.8-to-1, it’s 2-to-1. So, on average, players should win 30,000 rounds at $40 and lose 60,000 at $20. Action isn’t expected to break even in this situation, though, because bettors must pay the joints “juice” – a “vigorish” or “vig” – for the right to make this nominally “fair” wager. The vig on Buy bets at craps is 5 percent of the face value of the wager, rounded down to the next lower whole dollar. On $20, 5 percent is exactly $1. From $21 to $39, 5 percent is a dollar and change so rounding-down sets the vig at $1. Likewise, it’s $2 from $40 to $59, $3 from $60 to $79, $4 from $80 to $99, and so forth.

You can compare the averages the house nibbles from bettors’ bankrolls with Place and Buy bets by reducing the latter to percentages. As noted, Place bets on the four all come in at 6.67 percent. Superficially, $1 vig on a $20 Buy seems to be worth just $1/$20 or 5 percent for the bosses, and $1 on a $39 Buy appears to be even more attractive for players since $1/$39 is only 2.56 percent.

Buy bets are, in fact, superior to this. Were you to Place the four for $20 and win, you’d recover the $20 along with a $36 payoff so you’d be $36 ahead; if you lose, you’d be $20 behind. With these values, edge is [(3/9) x 36 - (6/9) x 20]/20, which equals -6.67 percent as anticipated.

To Buy a $20 four, you give the dealer $21, comprising $20 for the bet and $1 for the vig. Success gets you the $40 payoff along with your $20 bet, but the house keeps the $1 vig. Failure costs you the whole $21. Your net is therefore a $39 win or a $21 loss. The equivalent edge is accordingly [(3/9) x 39 - (6/9) x 21]/21 or -4.76 percent. Likewise, a $39 Buy requires you to set out $40. Victory earns you $78 and gets you back your $39 so your net gain is $77. Defeat puts you $40 behind. Equivalent edge is consequently [(3/9) x 77 - (6/9) x 40]/40, or -2.5 percent.

Equal $30 outlays show the contrast more directly. Placing this amount on the four would gain $54 or lose $30; the edge still works out to 6.7 percent. You have to squeeze the same $30 from your rack for a $29 Buy bet with $1 vig. You could bite the dust to the tune of $30 but could triumph with a $58 payoff and your $29 back, for a net of $57. As a result, effective edge is [(3/9) x 57 - (6/9) x 30]/30 or -3.3 percent – half that of the Place alternative.

An occasional enlightened establishment will pander to patrons by charging the vig only if a Buy bet wins. When you drop the dealer $30, it all goes on the number. Lose and this is what it costs. Win, and the payoff is $60 minus $1 vig plus your whole $30 back. So you’re putting up $30 to net $59, and the equivalent edge is [(3/9) x 59 - (6/9) x 30]/30 or -1.1 percent.

Nice. But not nearly as nice as the roughly half percent edge to which you can hold the casino by betting $5 on Pass or Come and taking $25 Odds. But, then, you’d be stuck with the point the shooter threw and not with the four, which your personal on-line haruspex predicted would be your lucky number for the day. Well, as the memorable muse, Sumner A Ingmark, metrified:
Some gamblers make their choices using erudition,
While others seek their fortunes based on superstition.