1. Consider the set B represented by the following list: ['water', 'coffee', 'tea', 'milk', 'juice', 'pop'].

a) Write the list as a proper set, including correct notation. (1 mark)

b) Give a new set A of cardinality 5 such that either AB or BA could represent a real set of objects. )

c) Find the cardinality of your set. Show all steps using proper notation.

d) Give a set C such that |C B|=3, or explain why this is impossible.

e) Give a set D such that |D B|=6, or explain why this is impossible.

f) Suppose P(n,r)=xC(2n,r-1) where n = 13, r = 5, and x is an integer. Find the value of x or prove that x must not be an integer. Show all steps.

2. Consider a game where you roll three dice that have 8 sides each. In this game, you might also draw cards from a deck of 52 cards.

a) Determine the theoretical probability for the product of the die rolls to be 4.

b) If you were to conduct an experiment and actually roll these dice, could you get an experimental probability result that is very different from the above? Explain in detail.

c) Find the probability of drawing a heart followed by a black card if the 1st card is a red 9, assuming cards are taken without replacement.

d) Show the probability distribution for a random variable X measuring the product of all your dice.

e) Find the expected value of X. (2 marks) f) Suppose you roll one of your dice and draw one card. Determine the probability that they match.