Figuring the Edge on A Parlay: How Do You Want to Look at It?

Parlays are serial bets in which players let winnings ride along with their original wagers. The idea is to catch a streak and progress, starting with small outlays. The approach is often used with longshots, low-probability wagers having high payoffs.

Parlays that come through can be spectacular. The nature of the beast is that they usually fail, of course. And they somehow seem to crash on what the hapless inevitably claim were their final rounds before they were gonna stop. Heart rending, despite the usual "well, it was only a buck."

The accompanying table shows the odds against winning and the earnings of single wagers and of two-, three-, and four-bet parlays starting with $1 on any number at double-zero roulette, and on yo-eleven at craps. Possible profits escalate sharply. But, as in all gambling against an opponent with an edge, the odds which must be overcome to win rise faster still.

Odds against winning and profits on roulette and craps parlays

	00 roulette		eleven at craps
wins	odds	profit	odds	profit
win 1	37-to-1	$35	17-to-1	$15
win 2	1,443-to-1	$1,295	323-to-1	$255
win 3	54,871-to-1	$46,655	5,831-to-1	$4,095
win 4	2,085,135-to-1	$1,679,615	104,975-to-1	$65,535

Don't start musing quite yet about that million-six for four somethings in a row at roulette, figuring the odds are steep but sooner or later it'll happen and why not sooner. It's one thing to bet $1 then let $36 or even $1,296 ride on the next two spins. But $45,656 gets beyond bragging rights about the one you missed.

Besides, table limits can keep you from betting $46,656 on the fourth round. It's not just the exposure; many joints would take the risk on high rollers with megabuck bankrolls and histories of average bets upwards of $25,000. Gaming honchos also accept the occasional jackpot on a low-limit slot or side bet. But not progressive table players who enter with peanuts and leave with the elephant. It has to do with a silly notion many casino bosses share with players, about solid citizens who had little to lose, but won big with the house's money rather than their own.

Technically, players can plan to parlay bets but quit at any stage, so intermediate payoffs are their money. They can grab it and head for the hills. But the matter isn't this simple. For example, it clouds the question of determining house advantage and is a contentious issue for many gaming aficionados.

Consider the yo bet at craps. Dice doyens know, or should know, that the casinos have an 11.1 percent edge on this proposition. For every dollar bet on the eleven, the house earns a theoretical 11.1 cents. The formula for those who think I just make this stuff up, is [(1/18)*15 - (17/18)*1]/1 because the chance of winning is 1/18, losing is 17/18, and a $1 bet pays $15.

What if you commit to a three-bet parlay on the yo? One way to figure the edge is that you're risking $1 to win $4,095. Odds of 5,831-to-1 mean you have 0.0171 percent probability of success. This yields an edge of [(0.000171)*4095 - (1 - 0.000171)*1]/1, or just under 30 percent. You bet $1, so the house would rate its theoretical earnings on your action as 30 percent of $1 or $0.30.

A different approach is that in three successful rounds, you bet $1 plus $16 plus $256, a total of $273. The house would calculate its theoretical take as 11.1 percent of the $273, or $30.33. This would be true for separate bets on each roll, spaced apart in time. You might also make a case that you risked your $1 three times at 11.1 percent -- so the house earns $0.333.

Are you giving the house $0.30 to risk $1 on a shot at $4,095? Is the casino somehow getting $30.33 to let you bet $273 trying for the same thing, when you only came to the table with $1? How about $0.333? And, does a three-stage commitment on the yo differ from a pull on a machine with three 18-position reels, where you either lose $1 or win $4,095 for a star in each window? Here's how the parlaying poet, Sumner A Ingmark, perceived the puzzle:

Some gamblers, eager to believe,
Will follow reasoning naive,
Their own good judgement to deceive.