Question

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

Answer 1

The correct answer is:\(=\frac{1}{3}\)
Here, X represents the number of sixes obtained when two dice are thrown simultaneously. Therefore, X can take the value of 0, 1, or 2.
∴ P (X = 0) = P (not getting six on any of the dice)\(=\frac{25}{36}\)
P (X = 1) = P (six on first die and no six on second die) + P (no six on first die and six on second die)
\(=2(\frac{1}{6}\times \frac{5}{6})=\frac{10}{36}\)
P (X = 2) = P (six on both the dice)\(=\frac{1}{36}\)
Therefore, the required probability distribution is as follows.

X	0	1	2
P(X)	\(\frac{25}{36}\)	\(\frac{10}{36}\)	\(\frac{1}{36}\)

Then, expectation of X\(=E(X)=\sum X_iP(X_i)\)
\(=0 \times \frac{25}{36}+1\times \frac{10}{36}+2\times \frac{1}{36}\)
\(=\frac{1}{3}\)