We assume that all birthdays are equally likely. As the births78 data set shows, this is not actually true. There are both weekly and seasonal fluctuations to the number of births.

a) Prove that if at least n birthdays have non-zero probability, then the probability of n matching birthdays is lowest when the birthdays are equally likely.

b) Use the births78 data set and simulations to estimate the probability that two (or more) people in a random group of n people have matching birthdays for several values of n. Assume that birthday frequencies are as they were in the United States in 1978. Further discussion of the birthday problem without the assumption of a uniform distribution of birthdays can be found in [Ber80].