>>Leadoff Prob of >= 1 run Avg. number of runs

>>-------------------------------------------------------------

>>single .428 .855

>>walk .432 .865

>I'm assuming that over eight years of data, the difference is statistically

>significant.

Over 8 years of data,

there have been (8 years)*(162 games/year/team)*(26 teams) = 33696 leadoff

innings. We are given an event x (e.g., x = leadoff single scoring).

Event x has a TRUE probability p of occurring each chance

(e.g. p = about 0.43). Using the DeMoivre-Laplace Theorem,

if we have n chances (in our case n=33696) we can compute the

probability of observing x between a fraction "a" and a fraction "b"

of all possible occurrences.

Assume that p = 0.43. Given 33696 possible occurrences, what is

the probability of observing leadoff singles scoring between fraction

"a" and fraction "b" of the time?

a b probability

---------------------------

0 0.428 22.9 %

0.428 0.429 12.6

0.429 0.430 14.5

0.430 0.431 14.5

0.431 0.432 12.6

0.432 1 22.9

So if the TRUE probability of a leadoff single scoring is 0.43, then

there is a 22.9 % chance that we will observe the leadoff single

scoring a fraction of 0.428 or less.

Similarly, if the true probability of a leadoff walk scoring is 0.43, then

there is a 22.9 % chance that we will observe the leadoff single

scoring a fraction of 0.432 or more.

There is therefore a (0.229)*(0.229) = 5.2 % chance that both leadoff

singles score 0.428 or less and leadoff walks score 0.432 or more.

Of course, there is also a 5.2 % chance that both leadoff

singles score 0.432 or more and leadoff walks score 0.428 or less.

RESULT:

If the probability of leadoff walks scoring is the same as the

probability of leadoff singles scoring, there is approximately

(5.2 + 5.2) = 10.4 % chance that we would observe a 0.004 (or more)

discrepancy over an 8-year observation period.

Draw your own conclusions.

----------

Dan Simon

Reference: A. Papoulis, "Probability, Random Variables, and

Stochastic Processes," Mc-Graw Hill Publishing, 1984, pages 42-49.