eBookProblem 4-23 (Algorithmic)Suppose that you are given a decision situation with three possible states of nature: S₁, S2, and S3. The prior probabilities are P(S₁) = 0.17, P(S₂) 0.53, and P(S3) - 0.30. With sample information I, P(I | S₁) = 0.13, PI | S₂) = 0.04, andP(I | S3)0.19. Compute the revised or posterior probabilities: P(S₁ | I), P(S₂ | I), and P(S3 | I). If required, round your answers to four decimal places.State of NatureS₁S253Total ProbabilityP (S₂11)PreviousNext

It is required to obtain the probabilities, , using the given prior probabilities of S1, S2, and S3,…

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A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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The prior probabilities for events A₁, A2, and A3 are P(A₁) = 0.20, P(A₂) = 0.30, and P(A3) = 0.50. The conditional probabilities of event B given A₁, A₂, and A3 are P(B | A₁) = 0.50, P(B | A₂) = 0.30, and P(B | A3) = 0.40. (Assume that A₁, A₂, and A3 are mutually exclusive events whose union is the entire sample space.) (a) Compute P(B ŉ A₁), P(B n A₂), and P(B n A3). P(B n A₁) P(B n A₂) = P(B n A3) = P(A₁)P(B | A₁) (b) Apply Bayes' theorem, P(A¡ | B) = 7 to compute the posterior probability P(A₂ | B). (Round your answer to two decimal P(A₁)P(B | A₁) + P(A₂)P(B | A₂) + + P(An)P(B | An)' places.) (c) Use the tabular approach to applying Bayes' theorem to compute P(A₁ | B), P(A₂ | B), and P(A3 | B). (Round your answers to two decimal places.) Events P(A₁) | P(B | A;) P(A¡n B) P(A¡ | B) A₁ 0.20 0.50 A₂ 0.30 0.30 A3 0.50 0.40 1.00 1.00 0.00
Suppose a person is thinking about entering a MarioKart competition at theirlocal videogame store. The competition costs 20 dollars to enter. The competition consists of 10 matches. The person knows that their chance of winning each match is 65% and that each match is independent of the others. The payout for the competition is: WIN ALL 10 Matches: 500 Dollar Prize WIN 9 Matches: 250 Dollar PrizeWIN 8 Matches: 50 Dollar Prize OTHERWISE: Nothing 1) Suppose that this person enters the competition 5 times.What is the probability they break even?(Hint: To break even they would need to win the $50 prize twice and lose the other three times. Use a modified binomial to calculate this probability.)
The following payoff table shows profit for a decision analysis problem with two decision alternatives and three states of nature. Decision Alternative = States of Nature 51 52 53 d₁ d₂ Suppose that the decision maker obtained the probabilities P(s₁) = 0.65, P(5₂) = 0.15, and P(53) = 0.20. Use the expected value approach to determine the optimal decision. EV(d₁) EV(d₂) = The optimal decision is? . 240 90 90 15 90 65
Imagine you have three coins: one is fair, one is biased (twice as likely to be heads as tails), the third is also biased (twice as likely to be tails as heads). You pick one of the three coins at random, and flip it 20 times. How should you decide which coin you are holding, based on the results, to maximize your probability of being correct?
PROBLEM (8) A risk averse decision maker with u(x) = square root(x) faces a lottery L that delivers $2500 or $100 with equal (1/2) probabilities. (a) Would he choose the lottery or a sure reward of $901 ?(b) Now suppose that he can buy “2 copies” of the lottery; this means there are 2 lotteries like above where the random prizes for each lottery is drawn independently with the probabilities above, and the sum reward from two lotteries is to be paid to the decision maker. Would the agent choose this “bundled lottery” or a sure reward of 2 x $901 = $1802 ? As you see, when the risk is independently replicated, the agent is less risk averse against aggregate risk that is composed of many “idiosyncratic risks”. PLEASE ANSWER ALL THE PART!
1. Two events A and B are such that P(A) = 0.5, P(B) = 0.3 and P(AUB) =0ff Calculate P(A B).
Let S = {E1, E2, E3} be the sample space of an experiment and let A = {E1}, B = {E2}, and C = events from S. The probabilities of the sample points are assigned as follows: P(E,) = 0.35, P(E2) = 0.30, and P(E3) = 0.35. Find P(BC). {E2, E3} be Select one: O a. 0.65 O b. 0.70 O c. 0.35 O d. 0.30
(a) In an experiment, a pair of dice (green and blue) are tossed and the numbers that come up are recorded. Assume that the first outcome is for the green die whereas the second outcome is for the blue die. (i) List the sample space S; (ii) List the elements for event X = {sum is greater than 8}; (iii) List the elements for event Y = {2 occurs on either die or both}; (iv) List the elements for event Z= {number greater than 4 comes up on the green die}; (v) List the elements corresponding to the event X Z; (vi) List the elements corresponding to the event Y N Z;
where the first outcome is one of X, Y, and Z, and the second outcome in each case is one of A and B. Use the following probabilities instead of those given in your text: Pr[X]=37 Pr[Y]=27Pr[Z]=27Pr[A|X]=45Pr[B|X]=15Pr[A|Y]=23Pr[B|Y]=13Pr[A|Z]=18Pr[B|Z]=78 Pr[(XorY)|B]=
Consider the following game in a game arcade. The probability of winning a game is p and hence loosingthe game is 1 − p. In order to win a prize you are given three choices:• A: win at least once in 6 games;• B: win at least twice in 12 games; and• C: win at least 3 times in 18 games.1. If p = 16 which of A, B, or C should you choose to maximize your probability of winning the prize?You are required to compute the probabilities for each option to justify your answer.2. Using the formulas derived in part (1), write a R-code to compute (not simulate) and plot the probabilityof winning the prize for options A, B, and C for different values of p varying from 0.1 to 0.9 inincrements of 0.1. Summarize your observation.Regarding the plot: i) All three curves should be drawn in the same plot, ii) Make sure that the threecurves are distinguishable (using different line styles, colors, and/or markers), and iii) make sure toadd a legend that identifies the curves for the three different options…

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