Negative Binomial Experiments

The 2004 baseball season was exciting since particular players had the opportunity to break single-season records. Let’s focus on Ichiro Susuki of the Seattle Mariners who had the opportunity to break the season record for the most hits that was set by George Sisler in 1920. Sisler’s record was 257 hits and Susuki had 255 hits before the Mariners’ game on September 30. Was it likely that Susuki would tie Sisler’s record during this particular game?

We can approximate this process as a coin-tossing experiment. When Susuki comes to bat, there are two relevant outcomes: either he will get a hit, or he will get an out. (We are ignoring other batting plays such as a walk or sacrifice bunt that don’t result in a hit or an out.) Assume the probability that he gets a hit on a single at-bat is p = .372 (his 2004 batting average) and we can assume (for simplicity) that the outcomes on different at-bats are independent.

Susuki needs two more hits to tie the record. How many at-bats will it take him to get two hits?

This is not a binomial experiment since the number of trials is not fixed. Instead the number of successes (hits) is fixed in advanced and the number of trials to achieve this is random. Consider

Y = number of at-bats to get two hits.

We are interested in probabilities about the number of bats Y.

It should be obvious that Y has be at least 2 (he needs at least 2 at-bats to get 2 hits), but Y could be 3, 4, 5, etc. Let’s find the probability that Y = 5.

First we know that the 2nd hit must have occurred in the fifth trial (since Y=5). Also we know that there must have been one hit and three outs in the first four trials – there are 4 C 1 ways of arranging the H’s and the O’s in these trials.

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Also the probability of each possible outcome is p^2 (1 - p)^3 , where p is the probability of a hit. So the probability that it takes 5 trials to observe 2 hits is

P(Y = 5) = (4 C 1) p ^2 (1 - .p)^3

Since p = .372 in this case, we get

P(Y = 5) = (4 C 1) (.372)^2 (1 - .372)^3

A general negative binomial experiment can be described as follows:

· We have a sequence of independent trials where each trial can be a success (S) or a failure.
· The probability of a success on a single trial is p.
· We continue the experiment until we observe r successes, and Y = number of trials we observe.

The probability that it takes us y trials to observe r successes is

P(Y=y) = (y -1 C r - 1) p^r (1 - p)^(y-r), y = r, r + 1, r + 2, ...

Let’s use this formula in our baseball example where r = 2 and p = .372. The table below gives the probabilities for the number of at-bats y = 2, 3, …, 9.

Y	P(y)
2	.1384
3	.1738
4	.1637
5	.1371
6	.1076
7	.0811
8	.0594
9	.0426

Note that it is most likely that Ichiro will only need three at-bats to get his two additional hits, but the probability of three at-bats is only 17%. Actually each of the values 2, 3, 4, 5, and 6 have probabilities exceeding 10%. There is a significant probability that Ichiro will take a large number of bats – by adding the probabilities in the table, we see that the probability that Y is at most 9 is .904, so the probability that Y exceeds 9 is 1 - .904 = .096.

For a negative binomial experiment where Y is the number of trials needed to observe r successes, one can show that the mean value is

E(Y) = r / p .

For our baseball example, r = 2 and p = .372, so the expected number of at-bats to get two hits would be E(Y) = 2/.372 = 5.4. It is interesting to note that although Y = 3 is the most probable value, Ichiro would average over 5 at-bats to get 2 hits in many repetitions of this random experiment.

Practice: Negative Binomial Experiments

Suppose a candy box contains a large number of candies of which 30% are peppermint. You select candies from the box with replacement until you choose a peppermint.

1. Explain why this is a negative binomial experiment and give values of r and p.

2. If Y represents the number of draws until you choose a peppermint, find the probability that Y is equal to 2.

3. Find P(Y <= 3).

4. How many draws do you expect to make before you choose a peppermint?

5. I simulated this experiment 20 times and I’ve listed the values of Y for these twenty experiments. Use this data to approximate P(Y <=3) and E(Y) and compare your answers with your answers to parts 3 and 4.

3 3 2 3 3 10 1 11 1 1 2 5 1 1 4 4 1 4 3 1