Question

Two dice are rolled. If each die has six faces which are numbered 2, 3, 5, 7, 11, 13, then the probability that sum of the numbers on the top faces being a prime number is

• A. $\frac{1}{6}$
• B. $\frac{5}{36}$
• C. $\frac{1}{18}$
• D. $\frac{1}{9}$
• E. $\frac{1}{12}$

Hint:

Logic Tip: The sum of two primes is almost never a prime because Odd + Odd = Even! The only way it's possible is if you use the only even prime number, $2$. This realization skips testing 25 combinations!

Answer 1

The Correct Option is A

Concept:
Probability is the ratio of favorable outcomes to the total number of possible outcomes. In this problem, all the faces on the dice are prime numbers. An important property of prime numbers is that all primes greater than 2 are odd. The sum of two odd numbers is always even (and thus not prime). The only way to get a prime sum from two prime numbers is to combine the only even prime (which is 2) with an odd prime.

Step 1: Determine the total number of possible outcomes.
Two 6-sided dice are rolled. $$\text{Total Outcomes} = 6 \times 6 = 36$$

Step 2: Analyze the parity of the faces.
The faces are $\{2, 3, 5, 7, 11, 13\}$. Notice that $2$ is an even number, and all other numbers $\{3, 5, 7, 11, 13\}$ are odd. If we roll two odd numbers, their sum is even (and $\ge 6$), so it cannot be prime. Therefore, a prime sum MUST involve the number 2.

Step 3: Test combinations involving the number 2.
Assume Die 1 rolls a '2'. We check the sum with all faces of Die 2: $2 + 2 = 4$ (Not prime) $2 + 3 = 5$ (Prime!) $2 + 5 = 7$ (Prime!) $2 + 7 = 9$ (Not prime) $2 + 11 = 13$ (Prime!) $2 + 13 = 15$ (Not prime)

Step 4: List the successful ordered pairs.
From Step 3, we found three valid sums if Die 1 is a 2. Pairs: $(2, 3), (2, 5), (2, 11)$. Because the dice are distinct, the reverse outcomes where Die 2 is a 2 are also valid: Pairs: $(3, 2), (5, 2), (11, 2)$. This gives exactly $3 + 3 = 6$ favorable outcomes.

Step 5: Calculate the final probability.
Divide the favorable outcomes by the total outcomes: $$P(\text{Prime Sum}) = \frac{\text{Favorable}}{\text{Total}} = \frac{6}{36}$$ Reduce the fraction: $$P = \frac{1}{6}$$ Hence the correct answer is (A) $\frac{1{6}$}.