Question

Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime?

$\left(\dfrac{1}{12}\right)^{12}$

$\left(\dfrac{1}{6}\right)^{12}$

$2\left(\dfrac{1}{6}\right)^{11}$

$\dfrac52\left(\dfrac{1}{6}\right)^{11}$

$\left(\dfrac{1}{6}\right)^{10}$

Answer: E

Concepts:dice (probability)prime multiplication principle

Difficulty rating: 1480

Small Hint:

A product of positive integers is prime only when one factor is a prime and the rest are $1$

Big Hint:

Choose which die shows the prime, and which prime; the other $11$ dice must show $1$

Solution:

The product is prime exactly when one die shows a prime ( $2,$ $3,$ or $5$ ) and the other eleven all show $1.$

The probability that any single die is the prime one is $\dfrac36 = \dfrac12,$ and each of the other eleven shows $1$ with probability $\dfrac16.$ Accounting for which of the twelve dice is prime, the probability is $\begin{aligned} 12 \cdot \dfrac12 \cdot \left(\dfrac16\right)^{11} &= 6 \cdot \left(\dfrac16\right)^{11} \\ &= \left(\dfrac16\right)^{10}. \end{aligned}$

Thus, E is the correct answer.

2005 AMC 10B Problem 12

Answer: E

Difficulty rating: 1480

Small Hint:

Big Hint:

Solution:

Problem 12 in Other Years