Baseball Musings
Baseball Musings
October 29, 2004
Hot Hand

The Sports Economist tries to use the Red Sox post-season win streak as an argument for team streakiness existing. I believe he makes a flaw in relating probability to outcomes in one area, however.


The conventional wisdom in the academic literature is that psychologists and behavioral economists have debunked the "myth of the hot hand," i.e. that players or teams are subject to streaky performance. Performance is just a random walk, they say. I don't buy it. The Red Sox were on a roll.

Note: The odds against LLL-WWWW-WWWW in a sequence of coin flips are 2047 to 1. Yes, Sox fans, deliverance from the curse of 1918 was a near miraculous event.


The problem is, any eleven game sequence that resulted in 8 wins for the Red Sox would have the exact same probability. So, if the LCS-WS went like this:

WLWLWWWWLWW

the odds of that would be 2047 to 1, but no one would consider it miraculous, since the Sox never trail in either of the two series. In fact, the odds of winning 8 out of 11, if the teams are evenly matched, is .113, a little better than 10%.

If you want to know the probability of the miracle, you have to start from where the Red Sox are down 3 games to 0. At that point, they have a 6.25% chance of winning the LCS. To win the championship, they have to win the LCS (6.25) and the Series (50%), so the chance of them winning everything at that point is 3.125%, or about 1 in 33. Those are low odds, and a comeback against them is certainly impressive. I don't know how many people would draw the miracle line there, however.

I also don't buy his "hot hand" argument. The Red Sox won 11 of 14 games played in the post season. The 95% confidence interval for a 14 game stretch is 3 to 11, so I'm not ready to reject the hypothesis that the Red Sox played teams that were evenly matched against them. I don't see any evidence that it was not a random occurance.


Posted by David Pinto at 10:01 AM | World Series | TrackBack (0)
Comments

Excellent point about the distribution of the wins.

I built a little simulator that simulated 100,000 ALCS series using the log5 formula and found the Red Sox coming back from 3-0 4.5% of the time. Those odds are a little less than your 6.25% I assume because the Yankees have a better chance of winning their home games than do the Red Sox using log5.

http://danagonistes.blogspot.com/2004/10/how-probable-was-bosox-comeback.html

Posted by: DanAgonistes at October 29, 2004 10:53 AM

I've had a chance to check my trusty stats book from days of yore (Walpole and Myers, 1985). The probability of having only 2 runs in a sequence of eleven trials, i.e. one run of Ls and one run of Ws, where either outcome is equally likely, is .012. That "p-value" is sufficiently low to reject the null hypothesis at the 95% confidence level, the standard used by David in his critique of my original post. Skeptics may wish to employ a higher standard of proof, but it's good enough for me. Indeed, Walpole and Myers employ a similar example from a twelve trial sequence, and state that "a sample containing only two runs is most unlikely to occur from a random selection process."

I continue to believe the Red Sox post-season performance was streaky.

Posted by: Skip at October 29, 2004 11:55 AM

What would you say was the probability of the Sox winning it all at the moment when they were headed to the bottom of the 9th in Game 4 of the ALCS, down 4-3 to the Yanks w/ Rivera coming in?

It's really just the same calculation you just did, except that we'd need to change the 0.5 probability of the Sox winning game 4 (which it presumably was before the game started) to something much smaller. That's the question then -- how often does a team win a game when Rivera is on the mound protecting a 1-run lead in the 9th?

My guess would be that the 0.5 becomes more like 0.05 (purely a guess), which makes the probability of winning it all (including the sweep in the Series) 1/10th of the 3.125%.

Fun stuff!

Posted by: Dan Moriarty at October 29, 2004 03:23 PM

Also Dan, if you factor in the Sox making eight errors and still winning the first two games against the Cards to keep the streak going, it gets silly.
I would conclude that it was destiny and destiny is always 100%

Posted by: steve g at October 30, 2004 09:26 AM

Short mathy discussion of probabilities with Rivera coming in @ Baseball Primer.

Posted by: Danil at October 31, 2004 10:04 AM

Does the 95% confidence factor in that it was impossible for the Red Sox to win 12, 13, or 14 of 14? The maximum games won was 11, but it those wins could have come in up to 19 games maximum. I'm not a probability person, it just doesn't seem right.

Posted by: Barron at November 1, 2004 12:09 AM