Problem 1 (3 + 3 + 3 = 9) Suppose you draw two cards from a deckof 52 cards without replacement. 1) What’s the probability thatboth of the cards are hearts? 2) What’s the probability thatexactly one of the cards are hearts? 3) What’s the probability thatnone of the cards are hearts?

Problem 2 (4) A factory produces 100 unit of a certain productand 5 of them are defective. If 3 units are picked at random thenwhat is the probability that none of them are defective?

Problem 3 (3+4=7) There are 3 bags each containing 100 marbles.Bag 1 has 75 red and 25 blue marbles. Bag 2 has 60 red and 40 bluemarbles. Bag 3 has 45 red and 55 blue marbles. Now a bag is chosenat random and a marble is also picked at random. 1) What is theprobability that the marble is blue? 2) What happens when the firstbag is chosen with probability 0.5 and other bags with equalprobability each?

Probem 4 (3+3+4=10) Before each class, I either drink a cup ofcoffee, a cup of tea, or a cup of water. The probability of coffeeis 0.7, the probability of tea is 0.2, and the probability of wateris 0.1. If I drink coffee, the probability that the lecture endsearly is 0.3. If I drink tea, the probability that the lecture endsearly is 0.2. If I drink water, the lecture never ends early. 1)What’s the probability that I drink tea and finish the lectureearly? 2) What’s the probability that I finish the lecture early?3) Given the lecture finishes early, what’s the probability I drankcoffee?

Problem 5 (4+4+4=12) We roll two fair 6-sided dice. Each one ofthe 36 possible outcomes is assumed to be equally likely. 1) Findthe probability that doubles were rolled. 2) Given that the rollresulted in a sum of 4 or less, find the conditional probabilitythat doubles were rolled. 3) Given that the two dice land ondifferent numbers, find the conditional probability that at leastone die is a 1. Problem 6 (8) For any events A, B, and C, prove thefollowing equality: P(B|A) P(C|A) = P(B|A ∩ C) P(C|A ∩ B)