A jar contains 3 red candies, 3 yellow candies, 3 green, 3 blue, and 3 orange. Amy, John, and Sandy take candies from the jar one after the other. First, Amy takes 3 candies. Then John takes 5 candies, and then Sandy takes 7 candies. A color is said to be "loved" if all candies of that color are eaten by the same person. Let X denote the number of loved candies. a) Find E(X) b) find V(X) c) Find P(X$\ge$4)
Not really sure how to proceed. Would it make sense to take a combinatoric approach and say, "there's 15 candies, I can arrange them all in a line in 15! ways"? Or would it be better to just compute the individual probabilities, such as the probability Amy takes 1/2/3 colors, John takes 2/3/4/5 colors, and Sandy takes....well I'm already confused so I guess that's not a good approach.
Use numbers $1,2,3,4,5$ instead of colors and let $B_i$ denote a random variable that takes value $1$ if candy $i$ is loved and takes value $0$ otherwise.
Then $X=\sum_{i=1}^5B_i$ .
Now apply linearity of expectation and symmetry to find $\mathbb EX$.
Further exploit: $$\mathsf{Var}(X)=\mathsf{Cov}(X,X)=\sum_{i=1}^5\sum_{j=1}^5\mathsf{Cov}(B_i,B_j)=20\mathsf{Cov}(B_1,B_2)+5\mathsf{Var}(B_1)$$
For finding $P(X\geq4)$ you can start with the observation that $X\geq4$ implies that Amy took $3$ candies of the same color.