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Expected length of longest winning streak in a 162-game baseball season

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ACG
#1
Apr29-09, 08:17 AM
P: 47
Hi!

The Red Sox just finished an 11-game winning streak, which made me wonder. Do most teams experience winning streaks this long over the course of a season? What is the expected value of the length of the longest winning streak for a .500 team over the course of a season?

In general: if you flip a coin N times with a probability of getting a head of P, what is the expected length of the longest series of consecutive heads embedded in the results of the N flips?

ACG
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CRGreathouse
#2
Apr29-09, 08:51 AM
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P: 3,684
To a good first approximation, there are (n - k + 1) independent chances with probability 1/2^k each to get a winning streak of length k in n games. (Of course they're not independent at all, but it's close enough.)

So the chance of getting an 11-game winning streak in 162 games is about 7%.
ACG
#3
Apr29-09, 09:06 AM
P: 47
Correct. However, as you know they're far from independent. If game 11 is a loss (50% chance) then there's no way for the set of games ranging from 2 to 11 to be the start of an 11-game streak. That dosen't sound like it will work.

ACG

CRGreathouse
#4
Apr29-09, 11:37 AM
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Expected length of longest winning streak in a 162-game baseball season

Quote Quote by ACG View Post
Correct. However, as you know they're far from independent. If game 11 is a loss (50% chance) then there's no way for the set of games ranging from 2 to 11 to be the start of an 11-game streak. That dosen't sound like it will work.
So in that case you overstate the probability of #2 to #11 by 1/2^11 (1/2^11 rather than 0). But if you know that the 11th game is a win, then you're understating the probability of #2 to #11 by 1/2^11 (1/2^11 rather than 1/2^10). So I don't think it's actually a problem!


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