Two dice are thrown simultaneously. If X denotes the number of fours, then the expectation of X will be:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{1}{3}\)
On each die, P(four) = 1/6 and P(not four) = 5/6. X can be 0, 1 or 2:
\( \displaystyle P(X=0) = \left(\frac{5}{6}\right)^2 = \frac{25}{36} \)
\( \displaystyle P(X=1) = 2\times\frac{1}{6}\times\frac{5}{6} = \frac{10}{36} \)
\( \displaystyle P(X=2) = \left(\frac{1}{6}\right)^2 = \frac{1}{36} \)
The expectation is \(E(X) = \sum x\,P(X=x)\):
\( \displaystyle E(X) = 0\times\frac{25}{36} + 1\times\frac{10}{36} + 2\times\frac{1}{36} \)
\( \displaystyle = \frac{12}{36} = \frac{1}{3} \)
Shortcut: X is a binomial variable with n = 2 and p = 1/6, so
\( \displaystyle E(X) = np = 2\times\frac{1}{6} = \frac{1}{3} \)