Saturday, December 23, 2006

84 vs. 56

Inspired by a recent post on the 2+2 gambling forums comparing Ted Williams' streak of reaching base in 84 consecutive games to Joe DiMaggio's famed 56, I'm doing a simple study comparing the relative difficulty of each man's achievement.

We'll consider a fictional star in the modern game who hits .330 and gets on base at a .430 clip, numbers similar to those Albert Pujols puts up annually. We'll assume our hero starts every game of his streak or takes days off without being used as a pinch hitter to make his task even more difficult. Our hero averages 3.83 at bats and 4.51 plate appearances per game, numbers that equate to 594 AB and 700 PA over 155 games. Each game follows a realistic distribution of opportunities: he typically gets 3 to 5 at-bats and 4 to 6 plate appearances, but there are frequent outliers. Specifically, the probability distributions look like this:

AB p PA p
0 0.01 3 0.04
1 0.02 4 0.6
2 0.06 5 0.22
3 0.21 6 0.1
4 0.5 7 0.03
5 0.15 8 0.01
6 0.04

7 0.01

I should note that these values are simply estimates. Anyone wishing to do a scientific study should compute the actual rates from game data.

Assuming the probability of getting a hit is fixed at .330 and that of getting on base fixed at .430, we can calculate the probability of each streak being extended in a given day. The simplest way to do this is to subtract from 1 the probability of the streak ending that day. The streak ends if our hero fails to get a hit--probability .670--in each at bat, for a total probability of .670^AB. Thus, our hero's hit streak is extended with probability 1-(.670^AB), an average of 76.1% of the time. Similarly, the on-base streak is extended with probability 1-(.570^PA), or 91.2% of the time.

If you're surprised that our hero is kept off the bases only 9% of the time, you're not alone. It's very rare for a strong on-base threat to get totally shut out.

Anyway, back to our study. The probability of starting a 56-game hit streak at any given time is .761^56, or .000000232, less than one in a million. It is an incredibly rare occurrence that mankind is lucky to have witnessed just once in the history of baseball.

What about starting an 84-game on-base streak? This happens with probability .000432. This is also a rare event, but it's 1858 times more likely than the 56-game streak. If the real Pujols is used in this pattern next year, there's a 6.47% chance he'll begin an 84-game on-base streak at some point in the year, and a .0036% chance he'll begin a 56-game hit streak. (If you're wondering, the probability of a .330 hitter batting .400 over 594 at bats is .018%, or around one in six thousand.)

In order for a .330 hitter to have a better chance of getting to 56 than 84, his OBP must be below .369. This means Ichiro is about equally likely to match either mark. Given that there is a natural limit of around .370 to any hitter's "true" batting average, but his "true" on-base percentage can go above .500 if pitchers simply stop throwing him strikes, we are far more likely to see an 84-game on-base streak in our lifetimes than we are to see DiMaggio matched or bested.

Conclusion: While getting on base may be more important than hitting for average, for a talented hitter a 56-game hitting streak is far more rare than an 84-game on-base streak.

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