Probability Expectation

Question

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 1/2302?

Answer 1

The number of integers needed to select a prime, $X$, is geometrically distributed with $p = 1/2302$. The probability of the event $\{X = k\}$ is equal to $\mathbb{P}(X = k) = (1-p)^{k-1}p$ for $k \geq 0$. Its mean is \begin{align} \mathbb{E}(X) = \sum_{k=0}^\infty k\mathbb{P}(X=k) = \frac{1}{p} \end{align} So for finding a prime number when drawing integers with 1000 digits we would expectedly need $2302$ draws.