# What is the probability that an element is in a subset of a set?

There's a list of numbers from 1 to 500 (both inclusive). Let's call this set $M$. I pick a subset $A$ uniformly at random from set $M$. The size of subset $A$ is $n$. Element $x$ is chosen uniformly at random from set $M$. What is the probability that $x$ is in set $A$?

• Wouldn't it just be $n/500$ assuming your elements in $A$ are distinct? – Alexis Olson Sep 20 '16 at 3:16

Assuming that $n$ is a fixed value, then let $\delta_x = 1$ if $x \in A$ and $\delta_x = 0$ otherwise. Since the set $A$ was selected through an equal probability selection of the elements of $M$ such that $\sum_{x \in M} \delta_x = n$, we can use the fact that $E(\delta_x) = P(x \in A)$ and take the expected value of that sum to show that $P(x \in A) = \frac{n}{500}$.