 For each pair (i, j) for k people in a room, we define the indicator random variable Xij, for formula_45, by

formula_46

formula_47

Let X be a random variable counting the pairs of individuals with the same birthday.formula_48

formula_49

For , if , the expected number of with the same birthday is formula_50 Therefore we can expect at least one matching pair with at least 28 people.An informal demonstration of the problem can be made from the list of Prime Ministers of Australia, of which there have been 29 , in which Paul Keating, the 24th prime minister, and Edmund Barton, the first prime minister, share the same birthday, 18 January.