Question

A dice is rolled three times and the sum of three numbers appearing on the uppermost dice is 15. The chance that the first roll was four is

• A. 2/5
• B. 1/5
• C. 1/6
• D. 2/4

Hint:

In conditional probability, first identify all outcomes that fit the given condition. This becomes your new, smaller sample space. Then, count how many of those outcomes also satisfy what the question is asking for.

Answer 1

The Correct Option is B

This is a conditional probability problem. The condition is that the sum of the three rolls is 15. Step 1: Find all possible outcomes that satisfy the condition (the sample space). We need to find all combinations of three dice rolls (x, y, z) where each number is between 1 and 6, and their sum is 15. Let's list them systematically:
Start with the highest possible first roll, 6: If the first roll is 6, the other two must sum to 9. The pairs are (3,6), (4,5), (5,4), (6,3). This gives us 4 outcomes: (6,3,6), (6,4,5), (6,5,4), (6,6,3).
If the first roll is 5: The other two must sum to 10. The pairs are (4,6), (5,5), (6,4). This gives us 3 outcomes: (5,4,6), (5,5,5), (5,6,4).
If the first roll is 4: The other two must sum to 11. The pairs are (5,6), (6,5). This gives us 2 outcomes: (4,5,6), (4,6,5).
If the first roll is 3: The other two must sum to 12. The only pair is (6,6). This gives us 1 outcome: (3,6,6). (A first roll of 1 or 2 is impossible, as the maximum sum of the other two dice is 12).

Step 2: Count the total number of possible outcomes. Total outcomes = 4 + 3 + 2 + 1 = 10.

Step 3: Find the number of favorable outcomes. The question asks for the chance that the "first roll was four". We need to count how many of our 10 possible outcomes start with a 4. Looking at our list, the favorable outcomes are: (4,5,6) and (4,6,5). There are 2 favorable outcomes.

Step 4: Calculate the probability. Probability = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes) = 2 / 10 = 1/5.