Multiple choice

A symmetrical die is thrown three times. If getting a six is considered to be a success, the probability of atleast two successes is

  1. $\dfrac{4}{27}$
  2. $\dfrac{3}{27}$
  3. $\dfrac{2}{27}$
  4. $\dfrac{1}{27}$
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C Correct answer
Explanation

Success probability p = 1/6, failure q = 5/6. For 3 trials, P(X >= 2) = P(X=2) + P(X=3). P(X=2) = 3C2 * (1/6)^2 * (5/6) = 3 * 1/36 * 5/6 = 15/216. P(X=3) = 3C3 * (1/6)^3 = 1/216. Total = 16/216 = 2/27.

AI explanation

The probability of getting a six on a die is 1/6. Using the binomial distribution for exactly two successes gives 3C2 times (1/6)^2 times (5/6)^1, which evaluates to 15/216. The probability for exactly three successes is 3C3 times (1/6)^3, which is 1/216. Adding these gives (15 + 1) / 216, which simplifies to 2/27.

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