Thursday, July 24, 2008

Binomial experiments

Today we talked about binomial experiments and the associated probabilities. Suppose we have a experiment where (1) you have a sequence of n trials, (2) on each trial, there are two outcomes, Success or Failure, (3) the chance of a Success (p) is the same from trial to trial and (4) the results of different trials are independent. Then the number of successes X has a binomial distribution.

Once you identify a binomial experiment, you have to figure out n, the number of trials, and p, the chance of a success. Then you can find probabilities of different outcomes by a table of binomial probabilities.

This can be applied to chance outcomes in baseball. In our Fathom lab, we suppose that Susuki has five plate appearances and we're interested in the number of times he gets on base. His OBP percentage is approximately 40%. This is (approximately) a binomial experiment with n = 5 and p = .5 where we define a success as getting on base. On Fathom, we can display the binomial probabilities -- we can find the chances that Susuki gets on base 0, 1, 2, 3, 4, or 5 times.

What is interesting is that actual baseball data (that is, the number of times Susuki gets on base different numbers of times) matches up well with the binomial distribution. The underlying assumptions aren't quite true. For example, the chance that Susuki gets on base likely changes depending on the pitcher and team that he faces. But this model gives reasonable answers and helps us understand the variation in the hitting data.

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