A fair die is rolled twice. The number that shows up after each roll is noted. Find the probability that the sum of the two numbers is a prime number or the product of the two numbers is a prime number.

To solve this problem, we need to understand the possible outcomes when a die is rolled twice.

Step 1: Identify the sample space A die has 6 faces, so when a die is rolled twice, the total number of outcomes is 6*6 = 36. These outcomes are all the possible pairs of numbers (1,1), (1,2), ..., (6,6).

Step 2: Identify the favorable outcomes We need to find the pairs where the sum or the product is a prime number. Prime numbers are numbers that have only two distinct positive divisors: 1 and the number itself. The prime numbers less than or equal to 12 (the maximum sum of two dice) are 2, 3, 5, 7, 11.

Step 3: Count the favorable outcomes By manually counting or using a systematic approach, we find that there are 27 pairs where the sum or the product is a prime number.

Step 4: Calculate the probability The probability is the ratio of the number of favorable outcomes to the total number of outcomes. So, the probability that the sum or the product of the numbers is a prime number is 27/36 = 0.75.