Our experiment consists of tossing a fair coin three times and observing a sequence (of length 3) of heads and tails. (1) List all possible outcomes comprising the sample space for this experiment: S = {...}. (2) Calculate the probability of the following events: A Exactly 2 tails in the experiment, B Head on the third toss in the last experiment (Experiment 3). C = There are even number of tails in the experiment Problem 2. Our experiment consists of tossing a fair die two times and observing a sequence (of length 2) of scores. (1) How many outcomes does the sample space consist of? (2) List all possible outcomes such that the multiplication of two scores is divisible by 4. (3) What is the probability of the sum of two scores being equal to 7? Problem 3. Let A and B be events with P(A) = 0.2 and P(B) = 0.3. What are the possible minimum and maximum values of P(AUB)? (You don't need to give a rigorous argument, but explain your answer). $1.2: METHODS OF ENUMERATION (BASIC COMBINATORICS) Problem 4. In a state lottery, four digits are drawn at random one at a time with replacement from 0 to 9. Suppose that you win if any permutation of your selected integers is drawn. Give the probability of winning if you select (1) 6, 7, 8, 9 (2) 7, 7, 8, 8 (3) 7, 8, 8, 8 Problem 5. (1) How many subsets of (1, 2, 3, 4, 5, 6, 7, 8) contain at least one of the elements of {1,2,3}? (2) How many subsets of {1,2,3,4,5,6,7,8} of size two (two elements) contain at least one of the elements of {1,2,3}? Problem 6. (1) Using the binomial expansion theorem we discussed in the class, show that (-1) () = 0. P=0 (2) Using the identy in part (a), argue that the number of subsets of a set with n elements that contain an even number of elements is the same as the number of subsets that contain an odd number of elements.