Suppose you take n people and for each of them the probabili

Suppose you take n people and for each of them, the probability that he was born at any given month is 1/12.

(i) Calculate p(n), the probability that some of them have the same month.

(ii) What is the minimal n such that p(n) > 50%?

Solution

Total number of people = n

Probability of each bornb in any given month = 1/12

(i) We can create an equation that will aid in finding the group:

P(A) = (12) * (11) * (10) * … (12 - n + 1) / 12^n

P(AC) = 1 - ((12) * (11) * (10) * … (12 - n + 1) ) / 12^n

Where

n = Number of people in the group, P(A) = Probability that no one shares a birthday, P(AC) = Probability that 2 people share the same birth month

(ii) Minimum n for P(AC) > 50% or 0.5

using the above equation,

for n=4, P(AC) = 0.42708 which is <0.5

for n=5, P(AC) = 0.61805 which is >0.5

Therefore, Minimum value of n=5 for P(AC) > 0.5 or 50%.