Example 417
An analyst predicted that 3.5% of all small corporations would file for bankruptcy in the coming year. For a random sample of 100 small corporations, estimate the probability that at least three will file for bankruptcy in the next year, assuming that the analyst's prediction is correct.
The distribution of the number X of filings for bankruptcy is binomial, with n = 100 and p — .035, so the mean of the distribution is fix — np = (100)0035) = 3.5
We will use the Poisson distribution, with mean A = 3.5, to approximate our binomial distribution. The probability function of the number X of bankruptcy filings is then approximated by
Thus, using e~3 5 = .030197 from Table 2 of the Appendix, we have
Therefore, the probability of at least three filings for bankruptcy from these 100 corporations is
(The reader is invited to speculate on the formidability of the task of computing this probability directly from the probability function of the binomial distribution. In fact, to three decimal places, this yields .684. Our approximation here is satisfactorily close to the exact probability.)
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