Express each of the following events in terms of the events A, B and C as...

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Express each of the following events in terms of the events A, B and C as well as the operations of complementation, union and intersection: (a) at least one of the events A, B, C occurs; (b) at most one of the events A, B, C occurs; (c) none of the events A, B, C occurs; (d) all three events A, B, C occur; (e) exactly one of the events A, B, C occurs; (f) events A and B occur, but not C; (g) either event A occurs or, if not, then B also does not occur. In each case draw the corresponding Venn diagrams. 2. You flip a fair coin 3 times, determine the probability of the below events. Assume all sequences are equally likely. (a) Three heads: HHH (b) The sequence head, tail, head: HTH (c) Any sequence with 2 heads and 1 tail (d) Any sequence where the number of heads is greater than or equal to the number of tails 3. Bob has a peculiar pair of four-sided dice. When he rolls the dice, the probability of any particular outcome is proportional to the sum of the results of each die. All outcomes that result in a particular sum are equally likely. (a) What is the probability of the sum being even? (b) What is the probability of Bob rolling a 2 and a 3, in any order? 4. Alice and Bob each choose at random a number in the interval [0, 2]. We assume a uniform probability law under which the probability of an event is proportional to its area. Consider the following events: A: The magnitude of the difference of the two numbers is greater than 1/3. B: At least one of the numbers is greater than 1/3. C: The two numbers are equal. D: Alice's number is greater than 1/3. Find the probabilities P(B), P(C), and P(AND). 5. Mike and John are playing a friendly game of darts where the dart board is a disk with radius of 10in. Whenever a dart falls within lin of the center, 50 points are scored. If the point of impact is between 1 and 3in from the center, 30 points are scored, if it is at a distance of 3 to 5in 20 points are scored and if it is further that 5in, 10 points are scored. Assume that both players are skilled enough to be able to throw the dart within the boundaries of the board. Mike can place the dart uniformly on the board (i.e., the probability of the dart falling in a given region is proportional to its area). (a) What is the probability that Mike scores 50 points on one throw? (b) What is the probability of him scoring 30 points on one throw? (c) John is right handed and is twice more likely to throw in the right half of the board than in the left half. Across each half, the dart falls uniformly in that region. Answer the previous questions for John's throw. 6. Prove that for any three events A, B and C, we have P(An BnC) ≥ P(A) + P(B) + P(C) -2. G1¹. Consider an experiment whose sample space is the real line. (a) Let {an} be an increasing sequence of numbers that converges to a and {bn} a decreasing sequence that converges to b. Show that lim P([a, b])=P([a, b]). Here, the notation [a, b] stands for the closed interval {r a ≤ x ≤ b}. Note: This result seems intuitively obvious. The issue is to derive it using the axioms of probability theory. (b) Let {an} be a decreasing sequence that converges to a and {n} an increasing sequence that converges to b. Is it true that lim P([an, bn]) = P([a,b])? 11-0 Note: You may use freely the results from the problems in the text in your proofs. 2 Express each of the following events in terms of the events A, B and C as well as the operations of complementation, union and intersection: (a) at least one of the events A, B, C occurs; (b) at most one of the events A, B, C occurs; (c) none of the events A, B, C occurs; (d) all three events A, B, C occur; (e) exactly one of the events A, B, C occurs; (f) events A and B occur, but not C; (g) either event A occurs or, if not, then B also does not occur. In each case draw the corresponding Venn diagrams. 2. You flip a fair coin 3 times, determine the probability of the below events. Assume all sequences are equally likely. (a) Three heads: HHH (b) The sequence head, tail, head: HTH (c) Any sequence with 2 heads and 1 tail (d) Any sequence where the number of heads is greater than or equal to the number of tails 3. Bob has a peculiar pair of four-sided dice. When he rolls the dice, the probability of any particular outcome is proportional to the sum of the results of each die. All outcomes that result in a particular sum are equally likely. (a) What is the probability of the sum being even? (b) What is the probability of Bob rolling a 2 and a 3, in any order? 4. Alice and Bob each choose at random a number in the interval [0, 2]. We assume a uniform probability law under which the probability of an event is proportional to its area. Consider the following events: A: The magnitude of the difference of the two numbers is greater than 1/3. B: At least one of the numbers is greater than 1/3. C: The two numbers are equal. D: Alice's number is greater than 1/3. Find the probabilities P(B), P(C), and P(AND). 5. Mike and John are playing a friendly game of darts where the dart board is a disk with radius of 10in. Whenever a dart falls within lin of the center, 50 points are scored. If the point of impact is between 1 and 3in from the center, 30 points are scored, if it is at a distance of 3 to 5in 20 points are scored and if it is further that 5in, 10 points are scored. Assume that both players are skilled enough to be able to throw the dart within the boundaries of the board. Mike can place the dart uniformly on the board (i.e., the probability of the dart falling in a given region is proportional to its area). (a) What is the probability that Mike scores 50 points on one throw? (b) What is the probability of him scoring 30 points on one throw? (c) John is right handed and is twice more likely to throw in the right half of the board than in the left half. Across each half, the dart falls uniformly in that region. Answer the previous questions for John's throw. 6. Prove that for any three events A, B and C, we have P(An BnC) ≥ P(A) + P(B) + P(C) -2. G1¹. Consider an experiment whose sample space is the real line. (a) Let {an} be an increasing sequence of numbers that converges to a and {bn} a decreasing sequence that converges to b. Show that lim P([a, b])=P([a, b]). Here, the notation [a, b] stands for the closed interval {r a ≤ x ≤ b}. Note: This result seems intuitively obvious. The issue is to derive it using the axioms of probability theory. (b) Let {an} be a decreasing sequence that converges to a and {n} an increasing sequence that converges to b. Is it true that lim P([an, bn]) = P([a,b])? 11-0 Note: You may use freely the results from the problems in the text in your proofs. 2