Question

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

Answer 1

Let X denote the number of sixes.

∴ Possible values of X are 0, 1, 2.

Let P(getting six when a die is thrown) = p = `(1)/(6)`

∴ q = 1 – p = `1 - (1)/(6) = (5)/(6)`

∴ P(X = 0) = P(no six) = qq = q² = `(5/6)^2 = (25)/(36)`

P(X = 1) = P(one six) = pq + qp = 2pq

= `2(1/6)(5/6)`

= `(10)/(36)`

P(X = 2) = P(two sixes) = pp = p² = `(1/6)^2 = (1)/(36)`
Expectation of X 
= E(X)

= \[\sum\limits_{i=1}^{3} x_i.\text{P}(x_i)\]

= `0 xx (25)/(36) + 1 xx (10)/(36) + 2 xx (1)/(36)`

= `(1)/(36)(0 + 10 + 2)`

= `(1)/(3)`.