Let A₁, A2, A3, ... be a sequence of events of a sample space. Prove that P(Ũ An) ≤ Ỹ P(An). n=1 n=1 This is called Boole's inequality.
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- Let S = {E1, E2, E3, EA} be the sample space of an experiment and let = {E1, E2}. B = {E2, E3}, and C = of the sample points are assigned as follows: P(E) P(E3) = 0.30, and P(E4) = 0.40. Find P(AU B). {Es, E,} be events from S. The probabilities 0.10. P(E2) = 0.20, A = %3D Select one: O a. 0.60 b. 0.20 O c. 0.90 O d. 0.30 e. 0.701aStarting at a particular time, each car entering an intersection is observed to see whether it turns left (L) or right (R) or goes straight ahead (S). The experiment terminates as soon as a car is observed to go straight. Suppose that the random variable y denotes the number of cars observed. (a) What are possible y-values? O all positive whole numbers O all integers O all real numbers greater than 1 all whole numbers greater than 1 O all positive real numbers (b) List five different outcomes and their associated y-values. Outcome Value of y RRS LLRLLS 6. LRRS 4 RRRS 4 S. 1 .
- Q6 At all times, an urn contains N balls- some white balls and some black balls. At each stage, a coin having probability p, 0 0} a Markov Chain? If so, explain why. Q6(ii.) Compute the transition probabilities P;,j: (Hint: At some stage, suppose there are i white balls (and N – i black balls). Think how you can get i or i – 1 or i +1 white balls in the next stage.) Q6(iii.) Let N = 2. Then the states are 0, 1, 2 and the transition probability matrix is given by (i) 2 2 2 2 Find the proportion of the time in each state.Let A and B be events in a sample space S. Show that if A and B are independent, then so are (a) A and B, (b) A and B, and (c) A and B.(3 points.) A bag of k∙n marbles contains n marbles of each of k distinct colors. A random sample (without replacement) of k marbles is drawn. Give an expression for Pk,n, defined as the probability the sample contains exactly two colors. Hint: there are (2) possible pairs of colors, and for each color there are (2²7) samples drawn from at most two colors, and this includes 2 (2) samples drawn exclusively from one or the other of the two colors, and of course there are (n) samples overall.
- A probability experiment is conducted in which the sample space of the experiment is S = {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16), event F = {9, 10, 11, 12, 13, 14), and event G = (13, 14, 15, 16). Assume that each outcome is equally likely. List the outcomes in F or G. Find P(F or G) by counting the number of outcomes in F or G. Determine P(F or G) using the general addition rule. List the outcomes in F or G. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. OA. For G={} (Use a comma to separate answers as needed.) B. For G = {}A probability experiment is conducted in which the sample space of the experiment is S= (6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17), event F = (10, 11, 12, 13, 14, 15), and event G = (14, 15, 16, 17). Assume that each outcome is equally likely. List the outcomes in F or G. Find P(F or G) by counting the number of outcomes in F or G. Determine P(F or G) using the general addition rule. List the outcomes in For G. Select the correct choice below and, if necessary, fillin the answer box to complete your choice. O A. For G= {10,11,12,13,14,15,16,17)} (Use a comma to separate answers as needed.) O B. For G={} Find P(F or G) by counting the number of outcomes in F or G. P(F or G) =D (Type an integer or a decimal rounded to three decimal places as needed.) Determine P(F or G) using the general addition rule. Select the correct choice below and fill in any answer boxes within your choice. (Type the terms of your expression in the same order as they appear in the original expression. Round to…:A coin is tossed four times, Find i. The probability of getting no heads 5/160 1/40 None of theseO 1/160 5/80 1. The probability of getting minimum of two tails 1/160 5/160 None of theseO 11/160 15/160 dal TOSHIBA
- Let 2 = {1,2, 3, 4, 5} be the sample space of a random experiment. Let also the events A = {1,2} and B = {2, 3, 4} be such that P(A N B) = 0.1, P(A) = 0.3 and P(B) = 0.8. Find the probabilities: (a) P({2}). (b) P({1}). (c) P(A°|B), (where Aº is the complement of A).Part 1: Suppose that F and X are events from a common sample space with P(F) # 0 and P(X) # 0. Prove that P(X) = P(X|F)P(F) + P(X|F)P(F). Hint: Explain why P(X|F)P(F) = P(Xn F) is another way of writing the definition of conditional probability, and then use that with the logic from the proof of Theorem 4.1.1. %3D Explain why P(F|X) = P(X|F)P(F)/P(X) is another way of stating Theorem 4.2.1 Bayes Theorem.Let S = {X1, X2, X3, X4, X5} be a sample space. a. List two simple events for S. b. List an event for S that is not simple. c. Assume S is an equiprobable space with probability function P(A). If Si is a simple event, then find P(Si). Also find P({X1, X3, X5}.
