Alternate Bets with Equal Edge May Perform Differently.

Picture a game like roulette but with 34 outcomes -- 0, 00, and 1 through 32. And think about two types of bets. Straight-up, on single numbers, paying 31-to-1; the likelihood of winning is 2.94 percent (one out of 34). Outside, for instance odd values between 1 and 32, paying 1-to-1 and losing on 0 or 00 as well as on even numbers; prospects of winning are 47.0 percent (16 out of 34). The house advantage works out to be 5.88 percent in either case.

Picture a game like roulette but with 34 outcomes -- 0, 00, and 1 through 32. And think about two types of bets. Straight-up, on single numbers, paying 31-to-1; the likelihood of winning is 2.94 percent (one out of 34). Outside, for instance odd values between 1 and 32, paying 1-to-1 and losing on 0 or 00 as well as on even numbers; prospects of winning are 47.0 percent (16 out of 34). The house advantage works out to be 5.88 percent in either case.

Fantasize that your favorite casino will book action down to $1 anywhere on the table. More, say you're a real sport and decide to risk $1, setting a goal of earning $31 or forfeiting your buck. And, make believe time is unimportant you'd as soon grind it out over a long session or do it in one fell swoop.

You accordingly devise three alternatives. There are other possibilities, but these seem to offer some good options. The first is to bet the dollar on a single spot, succeeding or failing on one spin. The second is to start with your money on Odd and, if you get lucky, parlaying your earnings; that is, plowing it all back up to five times, balancing profits of $1 + $2 + $4 + $8 + $16 = $31 against a wipe-out. The third is to grind away at a dollar a pop. Here, you can hit your goal in as few as 31 successive wins. Or, it may take a run of 32 wins and one loss, 33 wins and two losses, and so on -- providing that the losses don't knock you out before the wins get you to the top.

You'd start with the same dollar in each instance, looking for an identical profit, and making bets all with equal edge. Would your hopes for prevailing be equivalent as well, such that the only distinction would be the time factor?

The probability of victory betting one spin on a single number is simply the 2.94 percent or one out of 34. The parlay needs five wins on Odd in a row. That's 47 percent multiplied by itself five times, which comes out to 2.31 percent, just under one out of 43.

Betting a dollar per spin, you could amass the $31 by winning 31 rounds in a row. This is 47 percent multiplied by itself 31 times, or one out of less than 14 billion. Reaching $31 with this strategy through combinations of wins and losses greatly improves the chance of triumph, to roughly 0.3 percent or one out of 335.

The varying outlook for reaching the $31 profit starting with $1 partly results from the fact that the handle, the total amount wagered, differs among the methods. And the edge acts to shift a fraction of the handle from the solid citizens to the bosses. The penalty imposed by the edge on players therefore rises as the gross wager increases.

At 5.88 percent edge, $1 on a single number, one time, gives the house a theoretical cut of 5.88 cents. The parlay involves $1 on the first round -- 5.88 cents for the house; then $2 -- 11.76 cents for the house, followed by $4 at 23.52 cents, $8 at 47.04 cents, and $16 at 94.08 cents. The total is over $1.82. Betting $1 on each of 31 spins would also cost $1.82 in commissions to the casino. Improving the chance of winning by allowing some losses and greater numbers of wins would raise this amount. For instance, a series including 9 losses and 40 wins, a total of 49 spins, represents a theoretical $2.88 for the house.
There's another difference. Your goal might be to go all the way to $31 or bust out trying. Putting the dollar on a single number and taking one shot, there's no turning back. The parlay offers four occasions to change your mind and quit with an intermediate profit, but a single loss still sends you to the lockers. And betting $1 per spin offers multiple opportunities to skip with the joint's jack, along a broader range of stopping points; and depending where you are in the sequence, a loss or two doesn't necessarily spell curtains.

Had Aesop considered such trade-offs, the fable of the Tortoise and the Hare would have been far more complex than it is. For, as the sagacious scribbler, Sumner A Ingmark, wryly wrote:

Among a pragmatist's contentions:
Reality has more dimensions,
Than most philosophers' inventions.