and has asymptotic expansion: With days in a year, the average number of people required to find a pair with the same birthday is , somewhat more than 23, the number required for a 50% chance. In the best case, two people will suffice; at worst, the maximum possible number of people is needed; but on average, only 25 people are required An analysis using indicator random variables can provide a simpler but approximate analysis of this problem. For each pair (i, j) for k people in a room, we define the indicator random variable Xij, for , by Let X be a random variable counting the pairs of individuals with the same birthday. For , if , the expected number of people with the same birthday is ≈ 1.0356. 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. In the 2014 FIFA World Cup, each of the 32 squads had 23 players. An analysis of the official squad lists suggested that 16 squads had pairs of players sharing birthdays, and of these 5 squads had two pairs: Argentina, France, Iran, South Korea and Switzerland each had two pairs, and Australia, Bosnia and Herzegovina, Brazil, Cameroon, Colombia, Honduras, Netherlands, Nigeria, Russia, Spain and USA each with one pair. Voracek, Tran and Formann showed that the majority of people markedly overestimate the number of people that is necessary to achieve a given probability of people having the same birthday, and markedly underestimate the probability of people having the same birthday when a specific sample size is given. Further results showed that psychology students and women did better on the task than casino visitors/personnel or men, but were less confident about their estimates. Reverse problem The reverse problem is to find, for a fixed probability , the greatest for which the probability is smaller than the given , or the smallest for which the probability is greater than the given . Taking the above formula for , one has The following table gives some sample calculations. {| class="wikitable" |----- ! || ! || || || |----- | 0.01 | 0.14178 = 2.70864