Question

Among the positive integers less than $100,$ each of whose digits is a prime number, one is selected at random. What is the probability that the selected number is prime?

$\dfrac{8}{99}$

$\dfrac{2}{5}$

$\dfrac{9}{20}$

$\dfrac{1}{2}$

$\dfrac{9}{16}$

Answer: B

Concepts:prime basic probability systematic listing

Difficulty rating: 1420

Solution:

The available digits are $2,3,5,7$ . There are $4$ one-digit numbers and $4^2=16$ two-digit numbers, for $20$ total choices.

All $4$ one-digit choices are prime. A two-digit prime cannot end in $2$ or $5$ , so checking endings $3$ and $7$ gives the two-digit primes $23,37,53,73$ .

Thus $8$ of the $20$ choices are prime, and the probability is $\frac{8}{20}=\frac25$ .

Thus, the correct answer is B.

2015 AMC 10B Problem 11

Answer: B

Difficulty rating: 1420

Solution:

Problem 11 in Other Years