Question

A die is thrown twice and the sum of numbers appearing is observed to be 8. What is the conditional probability that the number 5 has appeared at least once?

• A. \( \frac{5}{36} \)
• B. \( \frac{2}{5} \)
• C. \( \frac{1}{18} \)
• D. \( \frac{1}{3} \)

Hint:

To find conditional probabilities, first determine the total number of possible outcomes that satisfy the given condition, and then determine the number of favorable outcomes. The conditional probability is the ratio of these two numbers.

Answer 1

The Correct Option is B

Let the two throws of the die be denoted by \( X_1 \) and \( X_2 \). We are given that the sum of the numbers appearing on the two dice is 8, i.e., \( X_1 + X_2 = 8 \). We are asked to find the conditional probability that at least one of the dice shows the number 5, given this condition. Step 1: Find the total number of possible outcomes where the sum is 8. The possible pairs of outcomes \( (X_1, X_2) \) such that \( X_1 + X_2 = 8 \) are: \[ (2, 6), (3, 5), (4, 4), (5, 3), (6, 2) \] Thus, there are 5 possible outcomes where the sum is 8. Step 2: Find the number of favorable outcomes where at least one die shows 5. Among the outcomes \( (2, 6), (3, 5), (4, 4), (5, 3), (6, 2) \), the pairs that include at least one 5 are: \[ (3, 5), (5, 3) \] Thus, there are 2 favorable outcomes where at least one die shows 5. Step 3: Calculate the conditional probability. The conditional probability is given by the ratio of favorable outcomes to total outcomes: \[ P(\text{at least one 5} \mid X_1 + X_2 = 8) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{2}{5} \] Thus, the conditional probability that at least one die shows 5, given that the sum is 8, is \( \frac{2}{5} \). Therefore, the correct answer is option (B)