Question

Two fair dice are rolled. Then the probability of getting a composite number as the sum of face values is equal to:

• A. $\frac{7}{12}$
• B. $\frac{5}{12}$
• C. $\frac{1}{12}$
• D. $\frac{3}{4}$
• E. $\frac{2}{3}$

Hint:

It's much faster to count the prime sums (only 5 values) and subtract from the total than to count all possible combinations for the 6 composite sums.

Answer 1

The Correct Option is A

Concept: When rolling two dice, there are $6 \times 6 = 36$ total outcomes. The sum of the faces can range from 2 to 12. We need to find the probability that the sum is a composite number (a positive integer greater than 1 that is not prime).

Step 1: Identify prime and composite sums between 2 and 12.

• Prime sums: $\{2, 3, 5, 7, 11\}$
• Composite sums: $\{4, 6, 8, 9, 10, 12\}$
• Neither (Unit): $\{ \}$ (1 is not a possible sum).

Step 2: Count the number of ways to get prime sums.

• Sum 2: (1,1) $\rightarrow$ 1 way
• Sum 3: (1,2), (2,1) $\rightarrow$ 2 ways
• Sum 5: (1,4), (4,1), (2,3), (3,2) $\rightarrow$ 4 ways
• Sum 7: (1,6), (6,1), (2,5), (5,2), (3,4), (4,3) $\rightarrow$ 6 ways
• Sum 11: (5,6), (6,5) $\rightarrow$ 2 ways Total ways for prime sum $= 1 + 2 + 4 + 6 + 2 = 15$.

Step 3: Calculate the ways for composite sums and the probability.
Total composite outcomes $= \text{Total} - \text{Prime} = 36 - 15 = 21$. \[ P(\text{Composite}) = \frac{21}{36} \] Simplify by dividing by 3: \[ P(\text{Composite}) = \frac{7}{12} \]