Question

An unbiased six sided die, with faces marked \(1\), \(2\), \(3\), \(4\), \(5\), 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 pairs of die faces that add up to each prime number between 2 and 12, then divide the count of favorable pairs by 36.

Answer 1

The Correct Option is C

Step 1: List all possible outcomes.
Each die has \(6\) faces, and the die is rolled twice, so the total number of equally likely outcomes is
\[ 6\times6=36 \]

Step 2: List the possible sums and identify the prime sums.
The sum of the two recorded numbers can range from \(2\) to \(12\). Among these, the prime sums are
\[ 2,\ 3,\ 5,\ 7,\ 11 \]

Step 3: Count the number of ways to get each prime sum.
Sum \(=2\): only \((1,1)\), so \(1\) way.
Sum \(=3\): \((1,2),(2,1)\), so \(2\) ways.
Sum \(=5\): \((1,4),(2,3),(3,2),(4,1)\), so \(4\) ways.
Sum \(=7\): \((1,6),(2,5),(3,4),(4,3),(5,2),(6,1)\), so \(6\) ways.
Sum \(=11\): \((5,6),(6,5)\), so \(2\) ways.

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

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

Final Answer:
The probability that the sum of the two recorded numbers is a prime number is
\[ \boxed{\dfrac{15}{36}} \]