Question

A fair six-faced dice, with the faces labelled ‘1’, ‘2’, ‘3’, ‘4’, ‘5’, and ‘6’, is rolled thrice. What is the probability of rolling ‘6’ exactly once?

• A. \( \frac{75}{216} \)
• B. \( \frac{1}{6} \)
• C. \( \frac{1}{18} \)
• D. \( \frac{25}{216} \)

Hint:

Use the binomial probability formula for independent events to calculate probabilities in dice or coin tossing problems.

Answer 1

The Correct Option is A

To determine the probability of rolling '6' exactly once when a six-faced die is rolled thrice, we can use the probability principles related to independent events and combinatorics.

1. Concept Understanding: When a die is rolled once, the probability of any specific outcome (e.g., rolling a '6') is \(\frac{1}{6}\). Conversely, the probability of not rolling a '6' (i.e., rolling '1', '2', '3', '4', or '5') is \(\frac{5}{6}\).
2. Rolling the Die Thrice: We need to calculate the probability of getting exactly one '6' in three dice rolls. This can occur in the following scenarios: (6, not 6, not 6), (not 6, 6, not 6), or (not 6, not 6, 6).
3. Probability Calculation for Each Scenario:
   • For the pattern (6, not 6, not 6), the probability is: \(\left(\frac{1}{6}\right) \times \left(\frac{5}{6}\right) \times \left(\frac{5}{6}\right)\)

   This can be computed as:

   • \(\frac{1}{6} \times \frac{25}{36} = \frac{25}{216}\)
4. Calculate Total Probability: Since there are three possible cases where the '6' can appear, we multiply the single scenario probability by 3:
   • Total probability = 3 times \(\frac{25}{216}\) = \(\frac{75}{216}\)
5. Conclusion: The probability of rolling a '6' exactly once in three dice rolls is \(\frac{75}{216}\), which matches the correct option given.