PLAYING IT SMART by ALAN KRIGMAN

When betting to climb out of a hole makes sense, and when it doesn't

Solid citizens who fall behind at the casino often keep playing in the hopes of recovering. Or, better, surging ahead.

This isn't necessarily bad. Strong gamblers size their bets and bankrolls according to the edge and volatility of a game, so they can ride through normal downswings. For example, a blackjack buff following Basic Strategy and betting $10 every round has 50 percent chance of sinking into a $100 hole at some point before finishing 160 rounds. Continuing with a stake greater than this would prevent a drop expected half the time from becoming fatal.

This doesn't mean that digging below the $100 loss guarantees recovery. Fortunes can worsen. Is persistence prudent? It depends on the dough players consider, before leaving home, to be a reasonable price to pay for the entertainment value of the casino experience and the shot at a financial windfall.

Chasing losses gets dangerous when players deplete the stakes they can rationally justify and risk money they can't. The high limits that purveyors of plasti-cash now extend on monetary advances, coupled with the ubiquity of the machines where these loans are dispensed, make this scenario particularly pernicious.

Reverse "utility" sometimes comes into play. Pretend you plunge down by $5,000 and don't know how you'll pay it back if you quit then and there. So you borrow another $2,500 to go on. You can't cover $5,000; inability to repay $7,500 seems hardly any worse.

In other cases, chasing losses is based on what's called the "law of the maturity of chances." This is the spurious supposition that there's a mechanism forcing the law of averages to work. Belief that as more bets lose, wins become increasingly due.

Pretend you learn that the chance of missing seven Reds in a row at single-zero roulette is under 1 percent. (To be precise, it's 0.9416 percent.) You write a column of figures on a napkin -- 10, 20, 40, 80, 160, 320, 640 -- and add them up to obtain $1,270. You have $1,300, enough to bet $10 on Red and double-up if you lose, for seven spins. If the first six are non-Reds, you'll be $630 down but can put up $640 to win $640 for a net $10 profit. You're not worried. You figure the chance of seven non-Reds in a row is under 1 percent, so you're 99 percent sure of success.

Your confidence is misplaced. After six -- or any number, for that matter -- of successive non-Reds, the wheel still has 18 Red grooves and 19 others. For all practical purposes, the grooves are identical and the ball is the same as on previous spins. So the chance of a Red outcome is 18 out of 37, which equals 48.65 percent. Nothing about the previous coups alters any of this.

It may also help to understand the significance of six successive non-Reds followed by another non-Red equaling 0.9416 percent. By geometry alone, the chance of a non-Red on any spin is 19 out of 37, or 51.35 percent. The likelihood of seven in a row is this fraction multiplied by itself seven times, or 0.9416 percent.

But what's the prospect of six non-Reds followed by a Red? It's 51.45 percent multiplied by itself six times then multiplied by the 48.65 percent probability of a Red. This equals 0.8919 percent. It's even less than that of seven non-Reds in a row.

Both of these probabilities are low because there are 128 unique ways that non-Reds and Reds can be arranged in seven spins. Each of these ways has a chance between 0.6449 percent (all Reds) to 0.9416 percent (all non-Reds). All of these ways, combined, yield the 100 percent. The issue is far more complex than simply the question of six non-Reds followed either by a non-Red or a Red.

Of course, good luck can strike. Like a jackpot on a slot machine after a long decline. Or a shooter at a cold craps table holding the dice for an hour, making point after point with lots of numbers in between. But if you elevate such happenstance into evidence that averages will out and falling fortunes surely rise, you haven't heeded this hortation of the poet, Sumner A Ingmark:

Examine theories you believe,
To spare yourself the need to grieve.

(c) 2008, ICON/Information Concepts Inc

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