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2.2.2 The birthday problem

Exercise 2.20.

Would you bet that at least two people in your workshop had exactly the same birthday? How large must a workshop be to make the probability of finding two people with the same birthday at least ${0.50}$ ?

Hint R code:

options(digits=2)
n = 3 ;  prod(366-c(1:n))/365^n  #test code
for (n in 20:25){
    p = prod(366-c(1:n))/365^n;
    print(c(n, p))
}

For ${2}$ people:

Ignoring leap years, there are ${365.365}$ possible pairs of birthdays, of which ${365.364}$ are different. Assuming all combinations are equally likely, probability both different is

\displaystyle\frac{365.364}{365.365}=\frac{364}{365}.

For ${n}$ students:

There are ${365^{n}}$ possible combinations of birthdays of which ${365.364.\ldots.(365-n+1)}$ are all different. Probability all different is

\displaystyle\frac{365.364.\ldots.(365-n+1)}{365^{n}}.

If ${n=22}$ , this probability is 0.52. If ${n=23}$ , this probability is 0.49.