Estimate the expected number of integers with 1000 digits that need to be selected at random to find a prime, if the probability a number with 1000 digits is prime is approximately 1/2302.

Question

Answer:2302

Answer 1

Let X_i denote the random variable for i^th selection being prime. 

We require expected value of sum of all X_i 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(∑ X_i) = ∑ E(X_i)

We have X_i  = 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(∑ X_i) = ∑ E(X_i) = 1

=> ∑ (1/2302) = 1

as summation is from 1 to n, we get

n/2302 = 1 => n = 2302

http://www.cse.iitd.ac.in/~mohanty/col106/Resources/linearity_expectation.pdf