The Quality of a Bet Depends on Your Viewpoint

Some friends have an ongoing argument about craps. Rosencrantz places the four for $5, looking for a $9 payoff. Gildenstern bets $5 in the field - hoping for a $10 payoff on rolls of two or 12, but happy with $5 return on three, four, nine, 10, or 11. Each thinks the other's is a sucker bet. "Which is better?" they asked me the other day. "Which is the lesser of the evils?" I corrected, then hastened to add, "It depends on your viewpoint."

House advantage or edge is the usual measure of bet quality. This can be found by taking the probability of winning times the win, subtracting the chance of losing times the loss, then dividing by the amount wagered. Here's how it works for the bets in question.

A $5 bet on the four can be decided nine ways. Three of the nine (1?3, 2-2, or 3-1 for a four) win $9. The other six of the nine (1-6, 2-5, 3?4, 4-3, 2-5, or 6-1 for a seven) lose $5. Edge is therefore 3/9 of the $9 win, minus 6/9 of the $5 loss, divided by the $5 bet. The math yields a negative 6.67 percent - the house's theoretical cut of everything placed on the four.

A $5 field bet can be decided 36 ways. Two of the 36 (1-1 or 6-6, for a two or 12) win $10. Fourteen of the 36 (1-2 or 2-1 for a three; 1-3, 2-2, or 3-1 for a four; 3-6, 4-5, 5-4, or 6-3 for a nine; 6-4, 5-5, or 4-6 for a 10; and 5-6 or 6-5 for an 11) win $5. The other 20 of the 36 each lose $5. Edge is therefore 2/36 of the $10 win, plus 14/36 of the $5 win, minus 20/36 of the $5 loss, divided by $5. The math yields a negative 5.56 percent - the house's theoretical cut of everything bet in the field.

"That proves it," exclaimed Gildenstern. "The house edge is less betting the field than placing the four. The field is better."

"Wait," I interjected. "There's another viewpoint for comparing place and field bets. Especially if you play long sessions and bet one or the other on every roll, except maybe the come-out."

The alternate quality measure isn't based on edge. It involves expected net after multiple throws, if the dice show a statistically-correct sampling of results. I used 36 rolls as an example.

After 36 rolls, $5 placed on the four is expected to win $9 three times - $27 in all. The bet is expected to lose $5 six times - $30 total. Overall, this is an expected net loss of $3.

Analogously, after 36 rolls, $5 in the field is expected to win $10 twice and $5 fourteen times -? $90 in all. The bet can be expected to lose $5 twenty times - $100 total. Overall, this is an expected net loss of $10.

"That proves it," exclaimed Rosencrantz. "Expected net loss is less placing the four than betting the field. Placing is better."

As I left, Rosencrantz and Gildenstern were deadlocked. Later, at a craps table, I weighed the wisdom of delineating data to deliberate from various viewpoints, versus just mocking sucker bets.

The relativistic approach was validated by what happened next. A high roller kept tossing $10 on hard six, parlaying when it won. A nearby know-it-all stage-whispered to his, ahem, daughter about "that jerk making sucker bets on the hardways." Up popped 3-3 - $90 payoff. The hardways winner left the $100 in action. The shooter bucked 3-3 right back - $900 payout. The player grabbed $500, leaving $500 on the hard six. Three rolls later - yep - 3-3. The bettor collected the $4,500 payoff and the $500 on the table, having already locked up $500. That's $5,490 profit on an initial $10 in five rolls. He colored-up, tipped the dealers $25, and started to leave. Abruptly, he returned and flipped the know-it-all's companion a green chip, saying, "Here honey, buy your old man a good book on how to win money at this game."

Snidely snicker at all sucker bets? Not in front of the solid citizens who win them. Or who aspire to do so. It brings to mind the musings of the prominent parlayer's poet, Sumner A Ingmark:

The pedants and redeemers,
By denigrating schemers,
Deprive the world of dreamers.