Question

An unbiased six-faced dice whose faces are marked with numbers 1, 2, 3, 4, 5, and 6 is rolled twice in succession and the number on the top face is recorded each time. The probability that the sum of the two recorded numbers is a prime number is ________

• A. \(\dfrac{3}{36}\)
• B. \(\dfrac{13}{36}\)
• C. \(\dfrac{15}{36}\)
• D. \(\dfrac{19}{36}\)

Hint:

List the possible sums from 2 to 12, keep only the prime ones, and count the ordered pairs that give each of those sums.

Answer 1

The Correct Option is C

Step 1: Set up the sample space.
The dice is rolled twice, and each roll can land on any of 6 faces, so the total number of equally likely outcomes is
\[ 6\times6=36 \]

Step 2: List the possible sums and pick out the prime ones.
The sum of the two rolls can be anywhere from 2 to 12. Among these, the prime numbers are 2, 3, 5, 7, and 11.

Step 3: Count the outcomes for each prime sum.
Sum = 2: only \((1,1)\), so 1 outcome.
Sum = 3: \((1,2),(2,1)\), so 2 outcomes.
Sum = 5: \((1,4),(4,1),(2,3),(3,2)\), so 4 outcomes.
Sum = 7: \((1,6),(6,1),(2,5),(5,2),(3,4),(4,3)\), so 6 outcomes.
Sum = 11: \((5,6),(6,5)\), so 2 outcomes.

Step 4: Add up the favourable outcomes.
\[ 1+2+4+6+2=15 \]

Step 5: Work out the probability.
\[ P(\text{sum is prime})=\frac{15}{36} \]

Final Answer:
\[ \boxed{\dfrac{15}{36}} \]