Question: 2017 Winter Final - 19
Let $n \geq 3$ be an integer and consider a group $P_1,P_2,\dots,P_n$ of $n$ people. Each of these
people has a uniformly random birthday, which is independent of the birthdays of the other people.
We ignore leap years; thus, the year has 365 days.
Define the random variable $X$ to be the number of unordered triples $\{P_i,P_j,P_k\}$ of people (i.e., subsets consisting of three people) that have the same birthday.
What is the expected value $\mathbb{E}(X)$ of $X$?
Hint: Use indicator random variables.
Define the random variable $X$ to be the number of unordered triples $\{P_i,P_j,P_k\}$ of people (i.e., subsets consisting of three people) that have the same birthday.
What is the expected value $\mathbb{E}(X)$ of $X$?
Hint: Use indicator random variables.
(a)
${\frac{1}{365^2}} \cdot n^3$
(b)
${\frac{1}{365^3}} \cdot n^3$
(c)
${\frac{1}{365^2}} \cdot {n \choose 3}$
(d)
${\frac{1}{365^3}} \cdot {n \choose 3}$