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What are the odds of that?

Posted by Doug on August 27, 2007

This post has nothing to do with football, unless you're the kind of person who sees football wherever you look. It was inspired by a question I saw on a fantasy football message board, and I might be able to make some loose football tie-ins at the end, but you may want to skip this one if you read this blog primarily for the football rather than the math.

Here is the inspiration: a post on the footballguys message board that said simply,

What are the odds of drawing the 14th pick out of 14 teams for three straight years?

The poster had apparently done just that. It was posted at 7:29 p.m. By 7:31, three people had posted the answer that most people would consider the right one.

(1/14)*(1/14)*(1/14) = 1/2744

Assuming everything is on the level, that guy's chances of drawing the fourteen slot in a given year would be 1 in 14. Since each year would be independent of the others, multiplying the yearly probabilities should give the overall probability of 1/2744. Fair enough. But the question "what are the odds of that?" can --- and often should --- be interpreted differently. [By the way, I am blurring, and will continue to blur, the distinction between odds and probability. Nothing bad will happen as a result.]

Suppose I flip a coin ten times and get: THHHTTHHTH. What are the odds of that?

Well, that depends on what you mean; it depends on what you think is the essence of a sequence of coin flips. The probability of that precise string is 1/2^10, or 1/1024. The probability of 6 heads and 4 tails is about 20.5% (that's 10-choose-6 divided by 2^10). The probability of a 6/4 split one way or the other is about 41%. So is THHHTTHHTH a rare event or not? It's a rare outcome, but all of the 1024 outcomes are rare. And we're often not interested in the outcome itself. Instead we group outcomes together into the events that we're interested in. We might be interested in the event "six of one, four of the other," which consists of a lot of different outcomes. That event is not particularly rare.

As a brief football-related aside, this is one reason --- not the only reason, but one reason --- to hate the expression

When you put the football in the air, three things can happen, and two of them are bad.

Actually, when you put the football in the air, there are about eleventy-bajillion different things that can happen. Grouping them into the three events: complete, incomplete, and interception, is arbitrary. If I was at my own 20-yard line, I could just as easily say, "when you put the football in the air, 75 different things can happen, and 72 of them are good." I could gain 80 yards, I could gain 79 yards, I could gain 78 yards, I could gain 77 yards, ..., I could gain 5 yards or less, I could throw an incompletion, or I could throw an interception.

Back to our fantasy football playing friend. If he had drawn the 6th pick for three straight years, or the 8th pick, or the 3rd pick, the same pick for three straight years, would we have heard from him? If so, then maybe the right question is "what are the odds of me getting the same draft slot for three straight years?" Answer: 1/14^2 = 1/196 (this was pointed out in the same thread, at 1:41 a.m.).

But as someone who wasn't involved, I might have a different perspective. If any single member of that league happened to draw the 14 slot all three years, you can bet I would have heard from him on the message board. Quite possibly if any person in that league had drawn the same slot for three years, we would have heard about it. But wait. This message board comprises people from many, many different leagues that probably have similar draws. From my perspective, maybe the question is "in a given year, what are the odds of me reading about some guy who got the same draft slot for each of the last three years?" And the probability of that is pretty high.

How high? Well, first, let's focus on a single 14-team league and examine the probability of somebody drawing the same slot (any slot) for three straight years. This turned out to be a hard question, at least for me. The answer is:

Sum_{k=1 to 14} P_k * Q_k

where P_k is the probability that there are exactly k guys with the same slot in the second year as the first, and Q_k is the probability that, given exactly k guys were in the same slot year one and year two, one or more of them would again land the same slot in year three. Even figuring P_k isn't an easy, but it turns out (google one of my favorite official mathematical words --- derangements --- for details) that

P_k = (1 / k!) * sum_{i=0 to (14-k)} (-1)^i / i!.

I think there must be an easier way to express Q_k, but what I came up with was:

Q_k = (1 / 14!) * sum_{j=1 to k} [(-1)^(j+1) * (k-choose-j) * (14-j)!].

Crunching the numbers, we find that the chance of somebody in a 14-team league getting the same slot three years in a row is about .06876. Roughly 7%. Now suppose that all the members of, say, 10 different leagues of the same kind frequent my message board. Then there is a 1 - (1-.06876)^10 probability --- about 51% --- that I, a random member of the message board, will be hearing about somebody who had a 1-in-2744 event happen to him.

This discussion is vaguely connected to my annual splits happen posts, where I point out highly non-interesting facts, like that Thomas Jones averaged nearly 10 fantasy points per game more against teams whose city name starts with A--M than against those that started with N--Z. A friend of mine read that post and remarked, "I don't even know how you thought to look for something like that." But the point is that I don't have to be clever. I don't have to know where to look. If you look at enough things, enough players' splits, enough fantasy draft drawings, you will see some things that seem absurdly unlikely.

This is one reason that it's not always appropriate to apply standard significance tests to facts you read about or see flashed on the screen during a game. You think Marty Schottenheimer is a crummy playoff coach. What are the chances that the collection of teams he's coached would, against the playoff opponent's they've faced, have a record of 5-13 or worse? You could make some assumptions and run the numbers; maybe you'd find that it is 4% or so. So you conclude: the probability of his record having happened due to chance is very, very low. Therefore, it probably isn't just chance; he must be a bad playoff coach.

That isn't necessarily appropriate. Why not? Because, even if all coaches were exactly average and chance was completely responsible for their records, there would undoubtedly still be coaches with records like Marty's (about 4% of all coaches, in fact!). You were (probably) checking the unlikelihood of Marty's record because you already knew it was bad. Do you see why that method is a self-fulfilling prophecy?

That's not to say that Marty isn't a bad playoff coach, or that statistical significance testing can never be used to examine such questions. It's just a reminder that "how unlikely is it that Marty's teams would go 5-13 in the postseason?" is a question that, just like "what are the chances of me getting the 14th slot in my draft three years in a row?" can be interpreted in different ways.

This entry was posted on Monday, August 27th, 2007 at 4:14 am and is filed under Statgeekery. You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.