What is the probability of k people having a coincidence of birthdays, taking leap-years into account? Hint: Consider all the days in a 4-year period, in which the leap-day 2/29 occurs once.
Help tonight/before tomorrow afternoon would be lovely.
In the course of 4 years (i.e. 3 regular years and 1 leap year), each regular day occurs 4 times while February 29 occurs once. There are 365 regular days. The total number of days in the course of 4 years is:
365 x 4 + 1 = 1461
During that period, a person celebrates his/her birthday 4 times if it falls on a regular day and 1 time if it falls on February 29 (of course, he/she can choose to celebrate it on February 28 or March 1, but that's not the point of the problem).
The probability that his/her birthday falls on January 1 is 4/1461. The probability is the same for each regular day. Meanwhile, the probability that his/her birthday falls on February 29 is 1/1461.
The probability that k people have their birthdays on January 1 is:
The probability is the same for the other 365 regular days. Meanwhile, the probability that k people have their birthdays on February 29 is:
Hence, the total probability that k people have the same birthdays is: