Alex throws a fair tetrahedral die numbered 1,2,3 and 4 and an octahedral (eight-sided) die numbered 1,2,3, …,8. He defines M as the product of this two numbers. Find: a) P (M is prime)

To find the probability that the product M of the numbers rolled on the fair tetrahedral and octahedral dice is prime, we can break down the problem into several steps. Step 1: Determine the sample space. The tetrahedral die has four sides with numbers 1, 2, 3, and 4, and the octahedral die has eight sides with numbers 1 through 8. So, the total number of outcomes in the sample space is 4 (for the tetrahedral die) times 8 (for the octahedral die), which is 4 * 8 = 32 possible outcomes. Step 2: Find the prime numbers in the product. To find the prime numbers in the product M, we'll first list all possible products by multiplying each number from the tetrahedral die by each number from the octahedral die. Then, we'll identify which products are prime. Possible products (M): 1 * 1 = 1 1 * 2 = 2 1 * 3 = 3 1 * 4 = 4 1 * 5 = 5 1 * 6 = 6 1 * 7 = 7 1 * 8 = 8 2 * 1 = 2 2 * 2 = 4 2 * 3 = 6 2 * 4 = 8 2 * 5 = 10 2 * 6 = 12 2 * 7 = 14 2 * 8 = 16 3 * 1 = 3 3 * 2 = 6 3 * 3 = 9 3 * 4 = 12 3 * 5 = 15 3 * 6 = 18 3 * 7 = 21 3 * 8 = 24 4 * 1 = 4 4 * 2 = 8 4 * 3 = 12 4 * 4 = 16 4 * 5 = 20 4 * 6 = 24 4 * 7 = 28 4 * 8 = 32 Now, let's identify the prime numbers in this list: 2, 2, 3, 3, 5, and 7. Step 3: Calculate the probability. To find the probability P(M is prime), we count how many favorable outcomes (prime products) we have and divide it by the total number of possible outcomes in the sample space. Probability (P) = Number of prime products / Total number of outcomes P = 6 / 32 = 3/16 So, the probability that the product M is prime is 3/16.