Question

If 12 fair dice are independently rolled, then the probability of obtaining at least two sixes is equal to __________ (round off to 2 decimal places)

Answer 1

Correct Answer: 0.6 - 0.65

1. Distribution of Successes: - Each die roll has a probability \( p = \frac{1}{6} \) of rolling a six. - The number of sixes in 12 rolls follows a binomial distribution \( X \sim \text{Bin}(12, \frac{1}{6}) \). 2. Probability of At Least Two Sixes: - The complement is the probability of fewer than 2 sixes: \[ P(X \geq 2) = 1 - P(X = 0) - P(X = 1). \] - Compute \( P(X = 0) \) and \( P(X = 1) \): \[ P(X = 0) = \binom{12}{0} \left(\frac{1}{6}\right)^0 \left(\frac{5}{6}\right)^{12} = \left(\frac{5}{6}\right)^{12} \approx 0.1122. \] \[ P(X = 1) = \binom{12}{1} \left(\frac{1}{6}\right)^1 \left(\frac{5}{6}\right)^{11} = 12 \cdot \frac{1}{6} \cdot \left(\frac{5}{6}\right)^{11} \approx 0.2681. \] - Total: \[ P(X \geq 2) = 1 - 0.1122 - 0.2681 \approx 0.6197. \]