Question

An urn contains 5 white and 10 black marbles. A fair die is rolled and that number ofmarbles is randomly selected (without replacement) from the urn.

Compute the expected number of black marbles.

Answer 1

Consider two random variables X and Y in which X be the outcome on die and Y be the number of black marbles.

Then, we need to find the expected number of black marbles.

That is, we need to find E(Y).

E(Y)= [P(X=1)*E(Y|X=1)]+ [P(X=2)*E(Y|X=2)]+ [P(X=3)*E(Y|X=3)]+ [P(X=4)*E(Y|X=4)]

+ [P(X=5)*E(Y|X=5)]+ [P(X=6)*E(Y|X=6)]

= [(1/6)*(1*10/15)]+ [(1/6)*(2*10/15)]+ [(1/6)*(3*10/15)]+ [(1/6)*(4*10/15)]

+ [(1/6)*(5*10/15)]+ [(1/6)*(6*10/15)]

= 0.1111+0.2222+0.3333+0.4444+0.5555+0.6666

= 2.3331.

the expected number of black marbles= 2.3331 2​​​​​​​