Consider two draws from a box with replacement contain 1 red ball and 3 blue balls. Let \(X\) be number of red balls. Let \(Y\) be 1 if the two balls are the same color and 0 otherwise. Let \(Z_i\) be the random variable which returns 1 if the \(i\)-th ball is red.

1. What is the sample space.
2. Write down the algebra of all events on this sample space.
3. What is the algebra of events generated by \(X\) ?
4. What is the algebra of events generated by \(Y\) ?
5. What is the algebra of events generated by \(Z_1\) ?
6. What is the algebra of events generated by \(Z_2\) ?
7. Which random variables are determined by an another of  the random variables. Why ? How is this reflected in the algebras ?
8. (*) What pair of random variables are independent ? How is this reflected in the algebras ?