A deck of 18 custom playing cards is numbered from 1 to 18. A card is drawn 6 times with replacement. If X is the number of draws that result in a prime number, what is the expected value of X (E(X))?

Free worked solution · Quant Blueprint

7/3 prime-numbered cards on average.

1. There are seven primes from one through eighteen: 2, 3, 5, 7, 11, 13 and 17. One is not prime.
2. For each draw define an indicator equal to one if the card is prime and zero otherwise. Its expectation is 7/18.
3. The total X is the sum of six indicators. Linearity of expectation gives E(X) = 6 × 7/18 = 7/3.
4. With uniform independent draws and replacement, X also has a binomial distribution with parameters 6 and 7/18.

Common mistake

An expected count need not be a whole number. It averages outcomes over repeated experiments.

Follow-up

Without replacement, the expected count is still 7/3 by linearity, but the draws are dependent and the distribution is hypergeometric rather than binomial.

Original educational explanation for this practice question. Reviewed 2026-10-13. Firm-style practice; the label describes the collection’s preparation context.