Question

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 approximately\(\frac{1}{{2302}}\)

Short answer

Answer:

The expected number of integers with 1000 digits that need to be selected at random to find a prime is 2302

Step-by-step solution

Note the given data

The probability a number with 1000 digits is prime is approximately \(\frac{1}{{2302}}\)

Definition and Theorem  

Definition : A random variable X has a geometric distribution with parameter p if

\(p\left( {X = k} \right) = {\left( {1 - p} \right)^{k - 1}}p\)where p is a real number with\(0 \le p \le 1\)

Theorem : If the random variable X has the geometric distribution with parameter p, then\(E\left( X \right) = \frac{1}{p}\)

Calculation

We randomly select integers until we find a prime.

Let X be the number of integers need to select a prime.

When we select n integers then the first (n-1) integers could not have been primes and then n^thinteger has to be prime.

Then by definition,\(p\left( {X = n} \right) = {\left( {1 - \frac{1}{{2302}}} \right)^{n - 1}}\left( {\frac{1}{{2302}}} \right)\)

The expected value of a random variable X with a geometric distribution is the reciprocal of the constant probability of success, p =P( prime) =\(\frac{1}{{2302}}\)

Then by theorem

\(\begin{array}{l}E\left( X \right) = \frac{1}{{\frac{1}{{2302}}}}\\\begin{array}{*{20}{c}}{}&{}&{}\end{array} = 2302\end{array}\)

Conclusion  

Thus , The expected number of integers with 1000 digits that need to be selected at random to find a prime is 2302