What Conditional Expectations Say About Betting the Odds at Craps

8 October 2012

By Alan Krigman

Many casino games proceed in stages. Bets on Pass at craps are illustrations. Prospects involve the chance of 1) winning or losing immediately – on the come-out roll – or establishing one or another of the box numbers as the point, and 2) subsequently winning or losing on the point. The chances combine to form the “joint probability” of success solid citizens face at the start of a coup. When the initial throw neither wins nor loses the but establishes a point, focus shifts to the “conditional probability” of winning if the number is repeated before a seven appears.

Probabilities are eight out of 36 to win even money with seven or 11 and four out of 36 to lose with two, three, or 12 on one roll. The other 24 out of 36 chances are ro establish points – three out of 36 for a four or 10, four out of 36 for a five or nine, and five out of 36 for a six or eight.

Players are sitting pretty on the come-out with twice as many ways to win as lose. If the wager isn’t resolved on the first throw and bettors could take it down, they’d have a huge edge in the game over the casino.. But the bets are considered contracts which must be left in action even though, after a point is established, edge shifts to the house. This, because the bet progressing through the come-out to the point still pays only even money, while the conditional probabilities of winning are adverse. In particular, fours and 10s have three paths to glory with the numbers and six to the doghouse with a seven so the conditional probability of winning on either of these points is 3/9 = 33.33 percent. Similarly, the conditional probability of success on fives and nines is 4/10 = 40.00 percent and on sixes and eights, it’s 5/11 = 45.45 percent.

The disadvantage on the point roll is tempered by two factors inherent in the game and one which players can control. The inherent factors are 1) the liability on the point is partially offset by the advantage coming out, and 2) the lower the conditional probability of a number to win, the less its likelihood of becoming the point. The third factor is that players can augment their original “flat” wagers during the point phase of the roll by taking Odds. This doesn’t affect the probability of winning. However, the additional money pays in proportion to the difficulty of success, thereby raising the return percentage on the gross sum at risk and lowering the house’s edge.

Make believe a table has a $10 minimum and offers triple (3X) Odds. Your strategy is to start the sequence with $10 on Pass and add $30 (three times the flat bet) when the point is established.

Say the point is a four or a 10. With no Odds, the conditional return is [(3/9) x $20]/$10 = 66.66 percent and the house edge is [(3/9) x $10 - (6/9) x $10]/$10 = - 33.33 percent. On fours and 10s, Odds pay 2-to-1. Therefore, with $10 flat and $30 Odds, payoff is $10 plus $60 and you recover your $40 for a $110 overall return. This is a [(3/9) x $110]/$40 = 91.67 percent return and a [(3/9) x $70 - (6/9) x $40]/40 = -8.33 percent house edge.

On points of five or nine, with no Odds, conditional return is [(4/10) x $20]/$10 = 80.00 percent and the house edge is [(4/10) x $10 - (6/10) x $10]/$10 = –20 percent. Odds pay 3-to-2 so $30 gets $45. The conditional return with 3X Odds is accordingly [(4/10 x $55)/$40 = 95.00 percent and house edge is [(4/10) x $55 - (6/10) x $40]/$40 = -5.00 percent.

When the point is six or eight, with no Odds, conditional return is [(5/11) x $20]/$10 = 90.91 percent and house edge [(5/11) x $10 - (6/11) x $10]/$10 = -9.09 percent. Payoff for the Odds is 6-to-5 so $30 gets $36. The $10 bet with triple Odds therefore wins $46 and gives back the original $40. Return is therefore [(5/11) x $86]/$40 = 97.73 percent and edge is [(5/11) x $46 - (6/11) x $40)/$40 = -2.27 percent.

Improvements in conditional return percentage and house edge increase as the Odds multiple gets larger. On a five, for instance, 10X Odds with$10 flat pays $160 and recovers $110. Return rises from 95 percent with 3X Odds to ((4/10) x $270)/$110 = 98.18 percent and edge falls from -2.27 percent with 3X Odds to [(4/10) x $170 - (6/10) x $110]/$110 = 1.82 percent.

This information underscores the wisdom of deciding the total you’re comfortable having at risk on a Pass bet and apportioning it by reducing the flat component and increasing Odds, consistent with table limits and your own loss tolerance. If you’re comfortable with a total of $50 on the line at a $10 table, $10 flat and $40 Odds would be statistically preferable to $20 flat and $30 Odds or $25 flat and $25 Odds. On the other hand, some craps aficionados find tables with exceptionally high Odds multiples – 10X or even $100X – and add more money then they can afford after the point is established. They do, indeed, get a high return percentage and low edge. But they’re vulnerable to being wiped out by normal swings in the game because they overbet their bankrolls. The inkslinger, Sumner A Ingmark, captured this conundrum with the couplet:
To optimize criteria, you must decide what’s scarier,
A profit less than you would like, or dealers saying “take a hike.”