Question

Two dice are thrown simultaneously. If \( X \) denotes the number of sixes, find the expectation of \( X \).

Hint:

For binomial trials, \( E(X) = np \); here, \( n = 2 \), \( p = \frac{1}{6} \).

Answer 1

\( X \): Number of sixes, \( X = 0, 1, 2 \).
Probability of six on one die: \( \frac{1}{6} \), not six: \( \frac{5}{6} \).
\[ P(X = 0) = \frac{5}{6} \cdot \frac{5}{6} = \frac{25}{36}, P(X = 1) = 2 \cdot \frac{1}{6} \cdot \frac{5}{6} = \frac{10}{36}, P(X = 2) = \frac{1}{6} \cdot \frac{1}{6} = \frac{1}{36}. \] \[ E(X) = 0 \cdot \frac{25}{36} + 1 \cdot \frac{10}{36} + 2 \cdot \frac{1}{36} = \frac{10 + 2}{36} = \frac{12}{36} = \frac{1}{3}. \] Answer: \( \frac{1}{3} \).