Question

# An urn contains three red balls, two blue balls and one yellow ball. Our experiment is...

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?

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

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