Question

A fair die is rolled three times. Let X denote the number of different faces showing, X = 1, 2, 3. Find E(X). Give a good explanation please

Answer 1

Total number of outcomes for three times rolled dice = 6³

Let X be the number of different faces.

P(X = 1) = P(all faces are same)

[ Possible events are { (1,1,1),(2,2,2) ....,(6,6,6) } = 6 outcomes ]

= 6 / 6³

= 1 / 36

P(X = 2) = P(2 different faces)

[ One number is fixed and other two numbers are different then

for first number there are 6 choices and second number there are 5 choices) ]

= ³C₂ * 1 * 6 * 5 / 6³

= 3 * 6 * 5 / 6³

= 15 / 36

P(3 differnt faces) = ³C₃ * 6 * 5 * 4 / 6³  

= 20 / 36

E(X) = X * P(X)

= 1 * 1/36 + 2 * 15/36 + 3 * 20/36

= 91/36 ( = 2.5278)