What good are bets on which you can lose more than you can win?

Most casino bets have payoffs that equal or exceed the amount in jeopardy on them. The majority of solid citizens not only prefer these wagers, they look for extreme cases. Who can resist a hundred thousand dollar payoff for a buck in a machine or on a bonus bet at a table? And, why not take a shot with chump change you can easily obtain, on a bundle you won't see any other way?

There's a class of wagers that do just the opposite. Wins pay less than losses cost. They're called "Lay bets." Are they strictly for suckers, or are there reasons why sophisticated gamblers may choose to include them in their bags of tricks?

Craps has a number of Lay bets. For example, with a Lay behind the four or 10, you can give the dealer $41 to win $20 and get back $40 of the $41 originally taken from your rack. Likewise, for a Lay on the five or nine, you may toss out $31 to pick up $20 and get back $30 of your $31. And the corresponding wager on the six or eight requires $25 to win $20 and recoup $24 of the $25. Odds on Don't Pass or Don't Come are similar. Pretend you bet $10 flat at a table that offers double Odds. To net $20 on the Odds, lay $40 on points of four or 10, $30 on points of five or nine, or $24 on points of six or eight.

The totals wagered are greater than the payoffs because you're more apt to enjoy ecstasy than suffer agony. Consider how this would work in a hypothetical "fair" game – one with no edge for either the house or the player. For this purpose, make believe a hat (a fedora would do the job) contains 10 marbles, colored either red or blue. You bet $10 and win on blue, lose on red.

Say the distribution of marbles is five red and five blue. You have a 50 percent chance of winning. When your marble is blue, the hatter pays $10 and returns your $10 bet. If a million people were to try, you'd expect 500,000 to win $10 and 500,000 to lose $10. Everything should come out even since it's a fair game.

Instead, imagine six marbles are red and four are blue. The chance you'll win is four out of 10 or 40 percent. How much is the payoff, remembering it's a fair game? Instead of figuring odds or probabilities per se, picture the million players again. The expectation is that 600,000 draw reds and lose. That's a $6,000,000 kitty. For balance, the 400,000 winners have their wagers returned and get paid $15 each because $15 x 400,000 is $6,000,000. This is like a typical casino bet. Put up $10 to earn $15, and be prepared for more defeats than victories.

Now make believe the hat holds two red and eight blue marbles. The chance of success is a strong eight out of 10 or 80 percent. You expect 200,000 of the million players to be vanquished, drawing red and losing $10, so the pot from which 800,000 victors are paid is $2,000,000. This means winners get their bets back plus $2.50 each because $2.50 x 800,000 is $2,000,000.

Casino games aren't fair, of course. There's a house advantage. So part of what would be a fair payoff goes to the bosses to fund their bonuses, the casino's mortgage and other bills, and the comps for all the shrimp you can peel yourself at the buffet you may think is free because you're such a great human being.

Craps isn't the only game in which you can make Lay bets. Banker at baccarat nominally pays even money but the house collects 5 percent commission; winners therefore actually get back $0.95 for every dollar at risk. Or analyze those roulette combination bets where you cover more than half the numbers. For instance, with $5 on each of two 12-number columns, the chances you'll win are 24 out of 38, over 63 percent, but you can lose $10 and only net $5.

Gamblers today tend to go for broke. So the most popular bets are those on which small sums up for grabs promise or at least tempt with rich rewards. (The laws of probability... what're they?) Lay bets are the opposite. If you're attracted by frequent small gains and can accept occasional big setbacks, they may work for you. As the sage songster, Sumner A Ingmark, saw the situation: