1. Each throw of a unfair die lands on each of the odd numbers 1, 3, 5 probability C and on each of the even numbers with probability 2C.

(a) Find C.

(b) Suppose that the die is tossed. Let X equal 1 if the result is an odd number, and let it be 0 otherwise. Also, let Y equal 1 if the result is a number greater than three and let it be 0 otherwise. Find the joint probability function of X and Y.

Suppose now that 12 independent tosses of the die are made.

(c) Find the probability that each of the six outcomes occurs exactly twice.

(d) Find the probability that 4 of the outcomes are either one or two, either three or four, and 4 are either five or six.

(e) Find the probability that at least 8 of the tosses land on even numbers.