Consider a random collection of n individuals Whenapproximat

Consider a random collection of n individuals. Whenapproximating the probability that no 3 of these individuals sharethe same birthday, a better Poisson approximation than thatobtained in the text (at least for values of n between 80and 90) is obtained by letting Ei be the event that thereare at least 3 birthdays on day i, i=1,......,365.

(a) Find P(Ei).

(b) Give an approximation for the probability that no threeindividuals share the same birthday.

(c) Evaluate the preceding when n=88 (which can beshown to be the smallest value of n for which theprobability exceeds 0.5).

Solution

(a) Find P(Ei).

Ei= =( 3/80) o (3/90) = 0.04 and 0.33

(b) Give an approximation for the probability that no threeindividuals share the same birthday.

P(x=3) = ((e^(-0.033))(0.033)^3)/ (3!) = 0.00579

(c) Evaluate the preceding when n=88 (which can beshown to be the smallest value of n for which theprobability exceeds 0.5).

Ei= =( 3/88) =0.0341