Here’s a simple probability problem:
My telephone rings 12 times each week, the calls being randomly distributed among the 7 days. What is the probability that I get at least one call each day?
This problem looks easy at first sight, and people are likely to jump to the following solution:
The denominator is the number of arrangements of 12 phone calls in 7 days (think it as distributing 12 indistinguishable balls into 7 bins) while the numerator
is the number of arrangements of 5 calls in 7 days assuming that there’s already one call each day. The above solution implies that each arrangement has equal probability; the problem, however, assumes that the arrangement is a multinomial distribution. More specifically, let
denote the number of calls received on the ith day of the week, then
where .
In order to solve the problem, we let denote the event that there is no call on the ith day, and the probability that there is at least one call each day is
We have
where is the probability that there is no call on i specific days. Therefore, the answer is
The above answer can be easily verified by a Monte Carlo simulation using the following MATLAB script:
n = 1e6; X = mnrnd(12, ones(1, 7) / 7, n); p = sum(all(X, 2)) / n;