1. (8 points) An experiment consists of flipping a coin and tossing one six-sided die. For example, one possible outcome is H3 (when the coin flip is heads and the die toss is 3). Let A be the event that the flip of the coin is T and B be the event that the toss of the die is 2 or less. (a) Find P(A). [1 exact] (b) Find P(B). [3 rounded] (c) Find P(A ∩ B). [3 rounded] (d) Find P(A ∪ B). [3 rounded] (e) Find P(A|B). [1 exact] (f) Find P(B|A). [3 rounded] (g) Are the events A and B mutually exclusive? Enter 0 if the answer is "no", or 1 if the answer is "yes". (h) Are the events A and B independent? Enter 0 if the answer is "no", or 1 if the answer is "yes".
2. (4 points) Two cards are selected without replacement from a standard deck of 52 cards. (a) How many kings are in the deck of 52 cards? [integer] (b) What is the probability that both cards are kings? [3 rounded] (c) What is the probability that the first card is a king and the second card is not a king? [3 rounded] (d) What is the probability that one of the cards is a king and the other card is not a king? [3 rounded]
3. (4 points) Suppose that A and B are independent events with P(A) = 0.4 and P(B) = 0.5. Find: (a) P(A ∩ B) [1 exact] (b) P(A ∪ B) [1 exact] (c) P(A|B) [1 exact] (d) P(B|A) [1 exact]
4. (4 points) Suppose that C and D are mutually exclusive events with P(C) = 0.4 and P(D) = 0.5. Find: (a) P(C ∪ D) [either 1 exact or integer] (b) P(C ∩ D) [either 1 exact or integer] (c) P(C|D) [either 1 exact or integer] (d) P(D|C) [either 1 exact or integer]
5. (4 points) A company makes gadgets using two machines, A and B. Machine A produces 70% of the gadgets, while machine B produces 30%. The proportion of defective gadgets is 3% for machine A, and 5% for machine B. (a) If a gadget is selected at random, what is the probability that it is defective? [3 exact] (b) If a gadget is defective, what is the probability that it was made on machine B? [3 rounded]
6. (4 points) A fair die is rolled three times. (a) What is the probability that the numbers obtained are the same? [3 rounded] (b) What is the probability that the numbers obtained are odd numbers? [3 exact] (c) What is the probability that the numbers obtained are all different? [3 rounded] (d) What is the probability that the numbers obtained are all different odd numbers? [3 rounded]