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If two standard dice are rolled, what is the probability that the sum and product of the numbers showing are prime?

Correct Answer:

**C**Try to find the smallest possible median.

- To find the events in which the sum and product of the numbers showing on the dice are both prime, we can first find those events in which the product is prime and then find in which of those cases the sum is also prime. For the next two steps we will consider only the products.
- Since a prime number has no positive factors except itself and 1, the only pairs of numbers on the dice that can multiply to a prime number are 1 and a prime number. For instance, if the first die shows 3, a prime number, and the second shows 1, the product is prime.
- There are three prime numbers on a standard die (2, 3, and 5). This means that if the first die comes up 1, there are three possible numbers (2, 3, and 5) that could make the product prime. Likewise, if the second die comes up 1, there are three numbers that could make the product prime. Because one number must be 1, these six events are the only ones in which the product is prime.
- We have identified the six events in which the product of the two numbers showing on the dice is prime. The pairs (with order) are (1,2), (1,3), (1,5), (2,1), (3,1), and (5,1). In which of these cases is the sum also prime? We can see that 1+2=3 is prime, but 1+3=4 and 1+5=6 are not, so only in the two cases in which the sum is 3 are both the sum and product prime.
- Each pair of numbers, including order, has an equal chance of coming up and there are 6*6=36 different pairs. Consequently, the probability that the sum and product are both prime is 2/36=1/18.