Question

If two dice are thrown, then the probability of getting co-prime numbers on the dice is:

• A. \( \frac{23}{36} \)
• B. \( \frac{13}{36} \)
• C. \( \frac{5}{6} \)
• D. \( \frac{1}{6} \)

Hint:

To determine the probability of rolling co-prime numbers on two dice, count the pairs where the greatest common divisor (GCD) is 1, then divide by the total number of outcomes (36).

Answer 1

The Correct Option is A

When rolling two dice, the total number of possible outcomes is: \[ 36 \quad \text{(since each die has 6 faces, so \( 6 \times 6 = 36 \))}. \] \

Step 1: Understanding Co-prime Numbers
Two numbers are considered co-prime if their greatest common divisor (GCD) is 1. Our task is to count all pairs of numbers (from the dice rolls) where the two values are co-prime. 

Step 2: Counting Co-prime Pairs
We list all valid pairs where the numbers are co-prime: If the first die shows 1: Every number is co-prime with 1, so there are 6 favorable outcomes.
If the first die shows 2: The numbers 1, 3, and 5 are co-prime with 2, giving 3 favorable outcomes.
If the first die shows 3: The numbers 1, 2, and 4 are co-prime with 3, giving 4 favorable outcomes.
If the first die shows 4: The numbers 1, 3, and 5 are co-prime with 4, giving 3 favorable outcomes.
If the first die shows 5: The numbers 1, 2, 3, and 4 are co-prime with 5, giving 4 favorable outcomes.
If the first die shows 6: The numbers 1 and 5 are co-prime with 6, giving 2 favorable outcomes. Summing these values: \[ 6 + 3 + 4 + 3 + 4 + 2 = 23 \] 

Step 3: Compute the Probability
The probability is calculated as the ratio of favorable outcomes (co-prime pairs) to total outcomes: \[ P(\text{co-prime}) = \frac{23}{36} \] Thus, the probability of rolling two numbers that are co-prime is: \[ \boxed{\frac{23}{36}}. \]