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August 25, 2005

Crooked Numbers

Going Streaking

by James Click

Not unlike the old Sports Illustrated Jinx, it seems that as soon as we talk about something here at BP, things turn around. Jonah Keri covered Sunday's A's game yesterday in his Game of the Week column, but it's safe to say that my last two columns--about the Royals' losing streak and the A's winning ways--have made large U-turns in the last week. The Royals' managed to finally break out of their near-record slump and it's this subject that deserves a little more of our attention.

Last week's column was a protracted discussion of the Royals' losing streak, its historical place, and a discussion of its likelihood. Unfortunately--as many readers and one fellow BP author pointed out--there was an error in the discussion, specifically this paragraph:

"Now, let's assume for a minute that the Royals are actually a .319 team (their current winning percentage). What are the odds that they'll lose any given 18 games in a row? By binomial distribution, we know that that probability is .000984 or approximately 1015.5:1. That seems very impressive, but that's only the probability that they'll lose any given stretch of 18 games. A 162-game season can be viewed as 144 separate 18-game opportunities to lose 18 games in a row. While the Royals chances of losing any given 18-games in 1015.5:1, their chances of losing any stretch of 18 games over the course of a 162 game season is actually closer to 6.6:1. What's more, the Royals, given their .319 winning percentage, had a 50:50 chance of losing 13 games in a row at some point during the season."

The probability of the Royals' losing any particular 18 games in a row is right, it is .000984. Furthermore, the odds of them losing 18 games in a row given 144 chances is actually 6.6:1. So if those are both right, where's the error? The problem comes from looking for streaks of exactly 18 games versus streaks of 18 or more. If a team loses 18 games in a row and then loses their next contest, they now have two 18-game losing streaks, overlapping by 17 games. Thus, if you view any losing streak of 18+n games as n+1 streaks of 18-games--as my calculations did--then you're vastly overestimating the likelihood that a given team will lose at least 18 games in a row.

As Rany Jazayerli pointed out, "In other words, it's not accurate to say that the odds of not losing 18 in a row on Day X is (1-.000984) = .999016, and (.999016)^144 = the odds of not losing 18 in a row over an entire season. On Opening Day, the odds of starting an 18-game losing streak is .000984; from that day on, the odds are (.000984) * (.319)." (He also pointed out that a 162-game season provides 145 opportunities to lose 18 games, not 144.)

A more accurate formula to answer the question we were asking--how likely is a team of a given winning percentage to lose a certain number of games in a row at some point during the season?--would be this: