11/12/09  What Are the Odds?

It's the middle of football season and we all know what that means. Not just BCS rankings or jostling for playoff positions, but what football means to fans coast to coast — gambling. There are many ways to bet on football, the favorite seems to be the point spread. This method is basically a way to handicap a game so each team has a roughly 50-50 chance of winning. Winning against the spread that is because neither the NCAA nor the NFL actually add or subtract these points from game scores.

Another favorite bet is the parlay, picking the winners in a series of games. This gets bettors a bigger payoff than a straight point-spread, 50-50 bet. Though it is harder to win as the more games in the parlay the longer the odds get. After all, there's only one winning combination in every parlay but more and more losing combinations the more games there are.

So I ask you, if a bookmaker offered 500-1 odds for a ten game parlay against the spread so each game is a toss-up, should you take it? Or is that a sucker's bet?

In order to know that we need to know how to calculate the odds of picking ten out of ten games right.

Let's begin with a simpler calculation, a 2 game parlay. Say team A is playing team B, and C is playing D. There are four possible outcome combinations, teams A and C win, teams A and D win, teams B and C win, teams B and D win. So you have one winning combination and three losing ones. The odds are 3-1 against your picking the parlay correctly.

Of course, trying to find all the possible combinations in a ten game parlay is cumbersome at best, so it'd be easier if we could use a mathematical formula. As we saw above with 2 games there are four combinations, 2 times 2 is four. In a four game parlay there are 16 combinations, 4 times 4 is 16. So then, should we just multiply the number of games times itself to get the odds? Would a ten game parlay have 100 combinations, 10 times 10? Should the odds be 99-1? Is our bookie giving us fantastic odds with that 500-1 payoff?

Actually no, because in a three game parlay there are 8 combinations, which isn't 3 times 3. A five game parlay gets you 32 combinations not 25, as you would with 5 times 5. What gives? Why does the formula work in two cases but not all the time?

Let's look at it from another angle with another kind of two contest parlay, horse racing's daily double. In this bet you must pick the winners of the first two races. Let's say there are ten horses in each race. This means there are ten possible winners in the first race and then ten possible winners in the second race. For each ten first race winners there are ten combinations with second race winners, so the total number of combinations for both races is 100. That's 10 times 10.

Now we can see the correct formula, it's not the number of races times itself, it's the number of possible winners in the first contest times the number of possible winners in the second. If you add a third contest you have to multiply the number of possible winners in the third race, too. If there were 10 horses in the third the odds of picking three straight races is 999-1. That's 10×10×10=1,000 combinations with one being a winner, so 999-1.

Calculating the odds of a parlay isn't an arithmetic progression, it's exponential. A two contest parlay is n (number of possible winners in first contest) times z (number of possible winners in second contest), or n×z. If you have the same number of contestants in each then n=z so you can replace z with n so the formula is n×n. To put that another way n squared, n to the power of 2, or n^2. Therefor, a ten contest parlay with an equal number of contestants is calculated as n×n×n×n×n×n×n×n×n×n, or n^10.

In football games there are only two possible winners in each game, so n=2. Which means a ten game parlay would calculate as 2^10, which equals 1,024. (That's 2×2×2×2×2×2×2×2×2×2 written out the long way.) Therefor the odds against winning a ten game parlay are 1,023-1.

Which means at 500-1 our bookmaker is not giving us terrific odds, but really bad odds. I've heard where sports books pay around 10-1 for a five game parlay. In such a bet the odds are not 10-1 or even 24-1 (5×5=25), but 31-1 (2^5=32). Now that's a sucker's bet.

If you think a ten game parlay is hard to hit, imagine trying to pick all 16 NFL games in a weekend right. The odds are 65,535-1. The odds of finding a bookie to take this bet are incalculable.