Question

An urn contains three red balls, two blue balls and one yellow ball. Our experiment is to draw a ball from an urn, replace it, and draw another. Define a random variable δ: Ω → R by δ(ω) = 1 if you draw the same color twice in a row, and δ(ω) = 0 otherwise. What is the expected value of δ?

Answer 1

3 red balls, 2 blue balls and 1 yellow ball.
First we find the probabilities from drawing balls from the urn.

Probability of drawing same color ball twice with replacement- P(Two red balls) + P(Two blue balls) + P(Two yellow balls)
P(Two red balls) = (first draw) x (second draw) = = 0.25
P(Two blue balls) = (first draw) x (second draw) = = 0.11
P(Two yellow balls) = (first draw) x (second draw) = = 0.028

Probability of drawing same color ball twice with replacement = 0.25 + 0.11 + 0.028 = 0.388
Probability of drawing different ball each time = 1 - 0.625 = 0.612