Product is Prime

Herbert rolls 6 fair six‐sided dice and computes the product of the numbers shown. What is the probability that this product is prime?

Answer

For the product of 6 dice to be prime, exactly one die must show a prime number (2, 3, or 5) and the remaining five dice must each show 1. Indeed, 1 does not affect the product (it contributes no prime factors), so having five 1s and one prime yields a product equal to that prime.

There are 3 possible primes on a given die: 2, 3, or 5.
There are 6 dice, so we can choose exactly one of them to be the “prime” die in $\binom{6}{1} = 6$ ways.

Hence, the total number of favorable outcomes is $6 \times 3 = 18.$ Since each of the $6^6 = 46{,}656$ outcomes is equally likely, the probability is

\frac{18}{6^6} \;=\; \frac{18}{46{,}656} \;=\; \frac{1}{2{,}592} \;\approx\; 0.0003858 \;\text{(about }0.0386\%\text{).}