hi,

this isProbability Theory

(1) A deck has eight cards: four red cards numbered 1 to 4, and four green cards numbered 1 to 4. First Brooke draws a card at random, and then Gautam draws one from the remaining seven. Let A be the event that Brooke's card has a higher number than Gautam's, and let B be the event that Gautam's card has a higher number than Brooke's. (a) Define a sample space for this experiment. Explain your notation. (b) Are A and B mutually exclusive? (c) Are A and B complements of one another? (d) What is A (as a subset of your sample space)? (e) What is A\" (as a subset of your sample space)? (2) Keep the same experiment from the previous problem. Let C be the event that Gautam's card is red. As subsets of your sample space... (a) What is AC? (b) What is A - C? (c) What is C - A? (d) What is (A 0)\"? (e) Are A and C mutually exclusive? (f) Are B and C mutually exclusive? (3) You have 3 quarters 4 dimes, two nickels, and 7 pennies in your pocket. You reach into your pocket and pull out a random selection of these coins. Describe the following events: (a) It is more than a dollar. (b) It is exactly 46 cents. (c) No two coins of the same type. (4) Let E, F and G be three events. Determine which of the following statements are correct and which are incorrect. Justify your answers. (For the true ones, you should show how the identity follows from the properties established in section 1.2 of the textbook or the elementwise method. For the falso ones, you should give an example where the two sides are different, explaining what each side is in your example.) (a) (EEF)UF=EUF (b) FCG U ECG = G(F U E)\" (c) (E U F)CG = ECFCG (d) EFUEGUFGCEUFUG. (5) Suppose that in the Buffalo metropolitan area, 60% of crimes occur in the city, and 80% occur at night. If 50% of all crimes occur in the city at night, what percentage occur outside of the city during the day? (6) Out of 435 UB math majors, 143 are in Actuarial Science, 100 are completing at least one other major, and 43 are in the Honors College. There are 13 in Actuarial science who are completing a second major, and of these 13, 11 are in the Honors College. The number who are in the Honors College or double majoring is 120. This includes those who are doing both. The number of Actuarial Science students in the honors college is 20. How many students are doing math as their only major, concentrating in something other than actuarial science and not in the Honors college. (7) Let S be a sample space and P a probability on events in S. For an event A, let Q(A) = P(A)2 and R(A) = P(A)/2. Is Q a probability? Is R? Give reasons why or why not? EXTRA PROBLEMS FOR THE MTH 511 STUDENTS (1) Let A and B be two events. Prove the following using the elementwise method. (a) (AAB)LJB =AUB (b) (A U B) AB = AB\" U A63. (2) Prove that B is impossible (i.e. B = o ) if and only if for every event A A: (BAC)U(BCUA) (3) Let {An};\"=1 be a sequence of events. Prove that for every event B, B( U A\") = U BA\". n=1 n=1 (4) Let {An};=1 be a sequence of events. Find a sequence {Bu};1 of mutually exclusive events such that for all n 2 1, n n U 141: = U B'' i=1 i=1 (5) Let {An};1 be a sequence of events. Prove that HE A.) g 2PM\")