A company with n employees publishes an internal calendar, where each day lists the employees having a birthday that day. What is the probability there is at least one day in a year when nobody has a birthday?

[tex]P(x)=^nC_xp^x(1-p)^x[/tex], where P(x) is the probability of getting success in x trials, n is the total number of trials and p is the probability of getting success in each trial.

We assume that the total number of days in a particular year are 365.

Then , the probability for each employee to have birthday on a certain day :

[tex]p=\dfrac{1}{365}[/tex]

Given : The number of employee in the company = n

Then, the probability there is at least one day in a year when nobody has a birthday is given by :-

[tex]P(x\geq1)=1-P(x<1)\\\\1-P(0)\\\\=1-(^nC_0(\frac{1}{365})^0(1-\frac{1}{365})^n)\\\\=1-(1)(\frac{364}{365})^n\ \ \ \ \ \ [\text{since}\ ^nC_0=1]\\\\=1-(\frac{364}{365})^n[/tex]

Hence, the probability there is at least one day in a year when nobody has a birthday =[tex]1-(\frac{364}{365})^n[/tex]