Playing it smart - Avoiding risk on advantage bets without conceding all the gains

Avoiding risk on advantage bets without conceding all the gains

Whew! Try to write that an advantage on a bet isn't the same as a guaranteed win. And that the laws of probability aren't the only factors gamblers might weigh, especially if the agony of defeat isn't worth the risk associated with the ecstasy of victory.

That was the intent of a recent article reviewing the theoretical cost the loss in expected value of taking down Don't Pass or Don't Come bets at craps after their points are established. I got reproved by outraged readers for everything from advising solid citizens to throw away money to substituting hunches for a sensible strategy and overlooking a better way to back off.

Mea culpa. They were right. For two principal reasons.

First, the discussion may not have strongly enough emphasized the distinction between merely guessing and letting utility outweigh probability at a particular juncture. That is, logically opting to value something like $10 in the rack over a 60 percent chance of sitting pretty with $20, when the downside is a 40 percent chance of having nothing. This, of course, would be the situation with $10 behind a five or nine after a bet on Don't Pass or Don't Come had survived the 8-to-3 disadvantage when coming-out.

Second, for Don't Pass and Don't Come bets during the point phase of a roll, there's another choice. It guarantees at least getting the wager back while offering a shot at earning a profit.

Make believe, for example, you've gone through the Don't Come with $10 and the point is nine. Leave it alone and you have six ways to win $10 versus four to lose $10. Or, if you wish, you may take down this advantage bet, recovering your $10 while neither winning nor losing. A third choice is to add a $10 Place bet to the nine. Now you have 60 percent chance to break even since a seven will win $10 on the Don't Come and lose $10 on the Place bet. However, you have 40 percent chance to net $4 because a nine will lose $10 on the Don't Come and win $14 on the Place bet.

Compare these options in terms of expected value. Letting the bet stand gives you 60 percent chance of finishing with $20 and 40 percent chance of finishing with zero. Expected value is 0.6 x $20 + 0.4 x $0, or $12. The $10 up for grabs is therefore worth a theoretical $12, which is why you're at an advantage.

Taking the bet down, you're 100 percent certain to finish with $10. No gain, no pain. You eliminate all the risk but it costs you the theoretical $2 extra you earned weathering the come-out.

Say you borrow $10 and Place it on the nine. Your chances of finishing are 60 percent with $10 and 40 percent with $14 (after you return the $10 loan, which was never vulnerable). Expected value is 0.6 x $10 + 0.4 x 14, or $11.60. You've avoided the risk but, rather than $2, you've only sacrificed a theoretical $0.40.

Similar math holds for insurance at blackjack. Pretend you bet $10 and get a blackjack but the dealer has ace-up. If you decline insurance and the dealer flips a 10, the chance of which is four out of 13, you'll push and get back the $10; with anything but 10 in the hole, the chance being nine out of 13, you recover the $10 and win $15. So expected value is (4/13) x $10 + (9/13) x 25, or $20.38. Insurance gets your $10 back plus a $10 payoff so the bet is worth a certain $20. Once your blackjack and the ace-up were exposed, your original $10 wasn't in peril either way. Insurance guaranteed $10 profit but it cost you a theoretical $0.38.

None of this should be taken as advice to give up part or all of the advantage earned getting to the stage of a round where the opening occurs. Rather, it's to highlight the statistical nature of conditional advantage. A bet will be profitable on the average but may lose as well as win in the here and now. Further, it's to show that situations may offer options to take a shot, mitigate risks, or guarantee a profit, and the trade-offs can be evaluated rationally. Like the message famous philosophers fathom from this memorable muse by the beloved bard, Sumner A Ingmark:

Explore alternatives most acutely,