Chapter 10: Brainteasers 361
Another one about birthdays
At a cinema the manager announces that a free ticket
will be given to the first person in the queue whose
birthday is the same as someone in line who has already
bought a ticket. You have the option of getting in line
at any position. Assuming that you don’t know anyone
else’s birthday, and that birthdays are uniformly dis-
tributed throughout a 365-day year, what position in line
gives you the best chance of being the first duplicate
birthday?
(Thanks to amit7ul.)
Solution
This is solved by an application of Bayes’ theorem.
Prob (A∩B)=Prob (A|B)Prob (B).You need to calculate two probabilities, first the proba-
bility of having the same birthday as someone ahead of
you in the queue given that none of them has a dupli-
cate birthday, and second the probability that none of
those ahead of you have duplicate birthdays. If there are
npeople ahead of you then we know from the previous
birthday problem that the second probability is
364!
(365−n)!365n−^1.The first probability is simplyn/365. So you want to
maximize
n364!
(365−n)!365n.This is shown as a function ofnbelow. It is maximized
whenn=19 so you should stand in the 20th place.
This maximizes your chances, but they are still small at
only 3.23%.