Question

Suppose n people are at a party. A) What is the probability that at least two people share a birthday? Hint: It is easier to first look at the complement of this event. Using a computer, create a plot showing this probability as a function of n.

B) How many people need to be in the room before this probability is greater than 1/2? Is this surprising?

C) Pigeonhole Principle. How many people need to be in the room before this probability is 1? Explain the intuition behind this.

Answer 1

a)

number of ways to select n different days out of 365 =³⁶⁵P_n =365*364*363*...(365-n+1)

number of ways to assing a birthday to n people =365ⁿ (As each person has 365 choices)

therefore P(at least two people share a birthday) =1-P(nne share as borthday)\

P(n) =1-365*364*363*...(365-n+1)/(365)ⁿ

below is probbaiility chart:

b)

here from above formulat and hit and trail ; number of people required so that probability is greater than 1/2 =23

this is surprising as within only 23 people probability increases that out of 365 days 2 people share same birthday.

c)

from pigeonhole Principle we require 366 people so that probability is 1 ; so that if 365 people all have different birthday ; that 366th person will share birthday with at least one 365 person as there are only 365 days,