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(A) The possible sums of two dice that are prime numbers are $2, 3, 5, 7, 11$.The sample space $S$ of pairs $(x, y)$ where $x+y$ is prime is:Sum $= 2:

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$\frac{8}{15}$

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(A) The possible sums of two dice that are prime numbers are $2, 3, 5, 7, 11$.The sample space $S$ of pairs $(x, y)$ where $x+y$ is prime is:Sum $= 2: (1, 1)$Sum $= 3: (1, 2), (2, 1)$Sum $= 5: (1, 4), (4, 1), (2, 3), (3, 2)$Sum $= 7: (1, 6), (6, 1), (2, 5), (5, 2), (3, 4), (4, 3)$Sum $= 11: (5, 6), (6, 5)$Total number of outcomes $n(S) = 1 + 2 + 4 + 6 + 2 = 15$.We need the probability that at least one number is a multiple of $3$ (i.e.,$3$ or $6$).The favorable outcomes are: $(2, 3), (3, 2), (1, 6), (6, 1), (3, 4), (4, 3), (5, 6), (6, 5)$.Number of favorable outcomes $n(E) = 8$.The required probability $P(E) = \frac{n(E)}{n(S)} = \frac{8}{15}$.

When $2$ dice are thrown, it is observed that the sum of the numbers appearing on the top faces of both dice is a prime number. What is the probability that at least one of the numbers in the pair is a multiple of $3$?

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