Multiple choice

A fair die is rolled repeatedly until a six is obtained. Let X denote the number of rolls required. The probability that $X = 3$ equals

  1. $\displaystyle \frac{25}{216}$
  2. $\displaystyle \frac{25}{36}$
  3. $\displaystyle \frac{5}{36}$
  4. $\displaystyle \frac{125}{216}$
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A Correct answer
Explanation

For X = 3, the first two rolls must not be 6 (probability 5/6 each) and the third must be 6 (probability 1/6). P(X=3) = (5/6) * (5/6) * (1/6) = 25 / 216.

AI explanation

This follows the geometric distribution, where we need the first two rolls to not be a six and the third roll to be a six. The probability of not rolling a six is 5/6, and the probability of rolling a six is 1/6. The combined probability is (5/6) * (5/6) * (1/6) = 25/216.

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