Question

1. Suppose there are n students in a class. Assume n ≤ 365, and all possible sequences of n birthdays are equally likely. We also assume that the birthday of a student is equally likely to be any one of the 365 days of the year (i.e., ignore leap years or seasonal variation in birth rate).

   (a) What is the probability that all n students have different birthdays?
   (b) Calculate this probability numerically when n = 23, 45, 65 and 70. Please feel free

   to use any computer software to calculate the numerical values.

Answer 1

a)probability that all n students have different birthday =P(selecting n different days from 365 days)

=³⁶⁵P_n/365ⁿ =365!/((365-n)!*365ⁿ)

b)

for n=23 ; probability =365!/((365-23)!*365²³)=0.492703

for n=45 ; probability =365!/((365-45)!*365⁴⁵)=0.059024

for n=65 ; probability =365!/((365-65)!*365⁶⁵)=0.002317

for n=70 ; probability =365!/((365-70)!*365⁷⁰)=0.000840