Two dice are thrown simultaneously. If X denotes the number of sixes, then the mean of X is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \displaystyle\frac{1}{3} \)
X can take the values 0, 1, 2. The probability of a six on one die is \( \frac{1}{6} \).
\( \displaystyle P(X=0)=\frac{5}{6}\cdot\frac{5}{6}=\frac{25}{36} \)
\( \displaystyle P(X=1)=2\cdot\frac{1}{6}\cdot\frac{5}{6}=\frac{10}{36} \)
\( \displaystyle P(X=2)=\frac{1}{6}\cdot\frac{1}{6}=\frac{1}{36} \)
\( \displaystyle E(X)=0\cdot\frac{25}{36}+1\cdot\frac{10}{36}+2\cdot\frac{1}{36} \)
\( \displaystyle =\frac{12}{36} \)
\( \displaystyle =\frac{1}{3} \)
Shortcut: X is binomial with n = 2 and \( p=\frac{1}{6} \), so \( E(X)=np=2\times\frac{1}{6}=\frac{1}{3} \).