## Estimate the probability that at least two people in a group of 30
## people have the same birthday.
##
## Assume that there are 365 days in every year, and that birthdays are
## independent and equally likely to occur on any day.


## The number of simulation replications.
nrep = 1e4

## The number of individuals in the group.
nsub = 30

## The number of simulations with at least one common birthday.
n1 = 0

## Simulation loop.
for (i in 1:nrep) {

  ## Generate the birthdays.
  bday = ceiling(runif(min=0, max=365, n=nsub))

  ## The unique birthdays.
  ubday = unique(bday)

  ## Check whether there were any shared birthdays.
  if (length(ubday) < nsub) { n1 = n1+1 }
}

## The estimated probability of a shared birthday.
p_est = n1 / nrep


## The exact probability.  The probability that all birthdays are different is:
##
## P(2nd bday is different from 1st) * P(3rd bday is different from first 2)
##  * ... * P(nsub^th bday is different from all others)
##
## We take 1 minus this to get the probability we want.
p_exact = 1
for (k in 1:(nsub-1)) {
  p_exact = p_exact * (1 - k/365)
}
p_exact = 1 - p_exact

## A numerical approximation to the exact probability.
p_approx = 1 - exp(-nsub*(nsub-1)/730)