Let Xi denote the random variable for ith selection being prime.
We require expected value of sum of all Xi to be 1. Linearity of expectation (refer link at bottom) says that expected value of sum of a random variable is equal to the sum of the individual expectations.
i.e., E(∑ Xi) = ∑ E(Xi)
We have Xi = 1/2302 and that is the same for all i. (once we take a number that number can be repeated also and hence events are independent)
So, E(∑ Xi) = ∑ E(Xi) = 1
=> ∑ (1/2302) = 1
as summation is from 1 to n, we get
n/2302 = 1 => n = 2302