What Will You Pay to Buy a Sure Thing?

Casino games have no obscure gambits letting solid citizens make single-stage bets, individually or combined, that either are sure to win or can't lose. Craps and blackjack, however, each have two-step bets offering this feature at the second level. Of course, these apparent gifts from the gambling gods have a hitch.

Here's the craps setup. Bet $12 - laying no odds - on Don't Pass or Don't Come. If the point becomes six or eight, make a Place bet on the same number for $12. On any other point, Place the same number for $10. Possible results are as follows:
* Six/eight - on a seven, Place bets lose $12 and Don't Pass bets win $12 for nets of zero; on the number, Don't Pass bets lose $12 and Place bets win $14 for $2 net gains.
* Five/nine - on a seven, Place bets lose $10 and Don't Pass bets win $12 for $2 net gains; on the number, Don't Pass bets lose $12 and Place bets win $14 for $2 net gains.
* Four/10 - on a seven, Place bets lose $10 and Don't Pass bets win $12 for $2 net gains; on the number, Don't Pass bets lose $12 and Place bets win $18 for $6 net gains.

The blackjack scheme is better known. When you get a blackjack and the dealer has ace-up, buy insurance for half your main bet.

* If the dealer has blackjack, your primary bet pushes and you win 2-to-1 on the half unit insurance; gain is 1 unit.
* If the dealer doesn't have blackjack, your primary bet wins 1.5-to-1 and the half unit insurance loses; gain is 1 unit.

What price do you pay for certainty under these circumstances? It's a sacrifice in the "expectation," the average amount you expect to win over a long time period.

For the craps situation, consider what happens after 990 decisions in each of the cases, anticipating the statistically-correct distribution of results in this many trials.
* Six/eight - the seven and the number should roll 540 and 450 times, respectively. A $12 Don't Pass bet, alone, should therefore win ($12 x 540) = $6,480 and lose ($12 x 450) = $5,400 for a net of $1,080. The two-stage hedge should win ($0 x 540) + ($2 x 450) = $900. You give up an expected $180, an average of over $0.18 per $12 Don't Pass bet.
* Five/nine - the seven and the number should roll 594 and 396 times, respectively. A $12 Don't Pass bet, alone, should therefore win ($12 x 594) = $7,128 and lose ($12 x 396) = $4,752 for a net of $2,376. The two-stage hedge should win ($2 x 594) + ($2 x 396) = $1,980. You give up an expected $396, an average of $0.40 per $12 Don't Pass bet.

* Four/10 - the seven and the number should roll 660 and 330 times, respectively. A $12 Don't Pass bet, alone, should therefore win ($12 x 660) = $7,920 and lose ($12 x 330) = $3,960 for a net of $3,960. The two-stage hedge should win ($2 x 660) + ($6 x 330) = $3,300. You give up an expected $660, an average of about $0.67 per $12 Don't Pass bet.

For the blackjack case, make believe the bet is $10. Consider what happens after 1,300 decisions in which you have a blackjack and the dealer shows an ace-up, assuming the statistically-correct distribution of results. Here, the dealer should and shouldn't have a 10 in the hole 400 and 900 times, respectively. Without insurance, you should push on 400 and win $15 on 900 for a net gain of $13,500. With insurance, you win $10 on all 1,300 for a net gain of $13,000. You give up an expected $500, an average exceeding $0.38 per $10 initial bet.

In these two-stage bets, you've beaten the odds to begin with, just to get into a favorable position. Is it better to take the sure thing, then and there? Or should you risk losing with a gamble offering a greater long-term expected profit? Choose for yourself. Decision scientists would call it a trade-off between utility and probability. And, while you're making up your mind, ponder the pertinent poetry of Sumner A Ingmark.

When you can't afford to lose it,
Take the safe way, don't excuse it;
When the gain outweighs the forfeit,
Go for the expected profit.