Suppose a circular clothes rack has n shirts, each of which has a uniformly chosen random color over a set of three possible colors (red, blue, or green). Assume all the colors are independent. Let x be the number of shirts that share a color with at least one of the two shirts adjacent to it. Find E[x] and Var(x). (Hint: Write x = ∑ᵢ=₁ⁿ yᵢ for some indicators yᵢ, and explicitly compute cov(yᵢ, yⱼ))

a) Conditional Probability
b) Joint Probability
c) Covariance Calculation
d) Probability Density