OT -- Probability Puzzle (100+ Point Prize With Thanks to Data)

[Ganchrow]

Thanks to Data for having generously chosen to forgo his justly earned prize for OT - 7/9/2010 Puzzle, those points (in addition to the book) have been rolled over.

This is a completely straightforward (although rather difficult) probability puzzle. Nothing up my sleeves this time ...

Inspired by this thoughtful assertion:

Take a team like NYY... They have 81 home games. We all can assume they will at least go 50-31 in those games.

Given:

1. 81 home games only, ignore away and postseason games
2. Same (constant) win probability for each home game
3. Prior win probability is Beta distributed

Prior assumptions:

1. You're 95% confident that the Yankees will win at least 50 home games.
2. You expect the Yankees to win 58 games

Q: If the Yankees lose their 1^st 10 home games, what's the probability they'll still win 50 or more home games over the season?

[tomcowley]

I probably botched something along the way, but I get 35.68%.

[Ganchrow]

I probably botched something along the way, but I get 35.68%.

Yep, that's exactly right. Rather quicker than I had expected. Well done.

Care to explain your answer for anyone following along at home?

(And additional thanks go out to Durito for having sweetened the pot. )

[Data]

I am not too good at this but I get 1%. Please explain.

[tomcowley]

Yep, that's exactly right. Rather quicker than I had expected. Well done.

Care to explain your answer for anyone following along at home?

(And additional thanks go out to Durito for having sweetened the pot. )

Sweet.

The first step is figuring out what the prior actually is. We know that the mean of the prior a/(a+b) =58/81 (since binomials are nice enough to "conserve" the mean to np). The other piece of information is that we are 95% sure of 50+% wins. So we have to figure out how a Beta prior translates to an 81-game binomial distribution. As it turns out, it follows the beta-binomial distribution http://en.wikipedia.org/wiki/Beta-binomial_model

So I made an excel sheet where I could input a (alpha), and it would calculate b (beta) from the fixed mean. Then I had one column of 0-81 wins and another column with the beta-binomial probability of actually getting that many wins given 81 games and a/b. I had a cell that summed the probability of 0-49 wins, and plugged in values of alpha until I converged on that probability being .05 (solver probably works, but I just did it by hand). Now we know what a and B are.

Then to condition on the 10 home losses we've seen, you just add 10 to b. Then you make a similar excel sheet and figure out the probability of 0-49 wins out of the 71 remaining games, and subtract from 1, which left 35.68%.

[Data]

Thank you, sir.

I am so not good at that that I got 2% but answered 1%. I never use these methods but do I want to? The team goes 0-10 and I should think that the team that I thought preseason is a 72% team is actually a 68% team? Thanks, but no thanks.

[tomcowley]

Yeah, the problem is the 95% assumption. It's basically the equivalent of watching the (guaranteed-identical-in-all-games) team go 72% over a full 162 games, then go on a losing streak. With that much data behind, it doesn't sound so bad. As an actual preseason projection, it's obviously absurdly tight.

[Ganchrow]

Did it almost exactly the same way, Tom.

And I obviously agree with you, too, Data ... I was just looking to quantify mizzoujohn's Yankees observation.

[RaginCajun]

U guys are nuts with math. I love math but have never done this much statistics. I took AP statistics in high school but never did something like this. I won't be taking statistics til my junior year of college. I'm a sophomore at Georgia Tech