A box of condiments has ten small packages, in which three are ketchup, three are mustard, and four are relish. A sample of three packages is randomly selected (without replacement) from the box.

a. Find the probability distribution for X, the number of mustard packages in the sample.

b. What are mean and variance of X?

c. What is the probability that at most one mustard package is selected?

d. What is the probability that at least one mustard package is selected?

e. What is the probability that X is within one standard deviation from its mean?

f. Now, suppose that you win $25 for each package of ketchup chosen and lose $15 otherwise. Let y denote the total winnings. (i) Find the probability distribution for y. (ii) Verify that this distribution satisfies the axioms of a probability distribution. (iii) Calculate the expected loss.