birthday problem.

Jul 2015
564
14
United States
How many randomly chosen guests should I invite to my party so that the probability of having a guest with same birthday as mine is at least 2/3?

How do we solve this?

Let X: denote guest having same birthday as mine.

P(X>=2/3)

we need to know how many n guests there will be so that P(X>=2/3)

So we would the correct way to solve this be: (1/365)*(364/365)^n-1 >= 2/3
 

romsek

MHF Helper
Nov 2013
6,647
2,994
California
$P[\text{someone has same birthday as you}] = 1 - P[\text{no one else has your birthday}] = 1 - \left(\dfrac{364}{365}\right)^n > \dfrac 2 3$

$\dfrac 1 3 > \left(\dfrac{364}{365}\right)^n$

$-\ln(3) > n \ln\left(\dfrac{364}{365}\right)$

$\ln(3) < n \ln\left(\dfrac{365}{364}\right)$

$\dfrac{\ln(3)}{\ln(365)-\ln(364)} < n$

$400.444 < n$

$n \geq 401$

Note this assumes the simple model of 365 days a year and does not account for leap years.
 
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