Question

Suppose that the birthdays of different people in a group of n people are independent, each equally likely to be on the 365 possible days. (Pretend there's no such thing as a leap day.)

What's the smallest n so that it's more likely than not that someone among the n people has the same birthday as you? (You're not part of this group.)

Answer 1

There are n individuals in the group whose birthday are equally likely and independent

let A is an event that no person has the same birthday as yours

P(A)= 1-P(A^c)

=1 - P(different birthday)

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

now we have to find the smallest value of n. By Birthday Paradox the smallest value of n is 23 which is not same as this problem.

suppose n=2 i.e there is two people and none of their birthday same as yours

then Q2 =1 -  (364/365)² ~ .28

if n=3 and none has same birthday as yours

Q3=1 - (365/365)*(364/365)*(363/365) ~ .55

and so on...

if there is n persons and none has same birthday as yours

Qn= 1- (356/365)*(364/365)*.....*(365-n+1/365)

if n is large then computations become harder so we use simple calculus

1 - x < e^-x

there is various method to find the value of n and one is

In order for the probability of at least one other person to share your birthday to exceed 50%, we need n large enough that

1 - (364/365) ^n-1 0.5

so we can find the value n for which n > 253

n=253 is the smallest number for which your birthday will be same as others.