The probability P(n) that an event characterized by a probability p occurs n times in N trials is given by the binomial distribution Consider a case where p << 1 and N >> 1.

Several approximations can then be made to reduce Eq. (1) to a simpler form. Using the result that In (1 - p) -p, show that (1 - p)^N_n e^-Np. Show that N!/(N A- n)! N^n. Use Stirling's approximation (Appendix A of Baierlein). Hence show that Eq. (1) reduces to W(n) lambda^n/n! e^-lambda where lambda = Np is the mean number of events.