A certain system can experience three different types of defects. Let A; (i = 1,2,3) denote the event that the system has a defect of type i. Suppose that the following probabilities are true.P(A₁) = 0.16P(A2) = 0.10 P(A3) = 0.07P(A₁ UA2) = 0.18P(A2 UA3) = 0.14P(A₁ UA3) = 0.18P(A1 A2 A3) = 0.02(a) What is the probability that the system does not have a type 1 defect?(b) What is the probability that the system has both type 1 and type 2 defects?(c) What is the probability that the system has both type 1 and type 2 defects but not a type 3 defect?(d) What is the probability that the system has at most two of these defects?

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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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A mouse is put inside a big room in which there are 4 exits through which the mouse can try to escape. There is a trap at each of the exits, e₁, e2, e3, e4, and the traps function properly with probabilities 0.3, 0.2, 0.3, and 0.5, respectively. Assume the mouse selects an exit at random. Let E₁ E (a) (b) = = the event that the mouse chooses the exit e; to escape, i = 1, 2, 3, 4. the event that the mouse escapes. Calculate the probability that the mouse escapes. It is known that the mouse already escaped, determine the exit that has the largest probability for the mouse to escape.
A machine produces defective parts with three different probabilities depending on its state of repair. If the machine is in good working order, it produces defective parts with the probability 0.02. If it's wearing down, it produces defective parts with probability 0.1. If it needs maintenance, a produces defective parts with probability 0.3. The probability that the machine is in good working order is 0.8. The probability that it needs maintenance is 0.1. Compute the probability that a randomly selected part will be defective.
If an experiment is conducted, one and only one of three mutually exclusive events S1, S2, and S3 can occur, with these probabilities: P(S1)=0.2, P(S2)=0.5, P(S3)=0.3. The probabilities of a fourth event A occurring, given that event S1, S2, or S3 occurs, are P(A|S1)=0.02, P(A|S2)=0.01, P(A|S3)=0.03. Find P(S3|A)?
A certain system can experience three different types of defects. Let Ai (i = 1,2,3) denote the event that the system has a defect of type i. Suppose that the following probabilities are true. P(A1) = 0.10 P(A2) = 0.07 P(A3) = 0.06P(A1 ∪ A2) = 0.11 P(A1 ∪ A3) = 0.13P(A2 ∪ A3) = 0.11 P(A1 ∩ A2 ∩ A3) = 0.01 (c) Given that the system has at least one type of defect, what is the probability that it has exactly one type of defect? (Round your answer to four decimal places.) (d) Given that the system has both of the first two types of defects, what is the probability that it does not have the third type of defect? (Round your answer to four decimal places.)
A certain system can experience three different types of defects. Let A, (i = 1,2,3) denote the event that the system has a defect of type i. Suppose that the following probabilities are true. P(A1) = 0.15 P(A2) = 0.09 P(A3) = = 0.07 P(A₁ U A₂) = 0.16 P(A₂ U A3) = 0.13 P(A₁ U A3) = 0.17 P(A1 A2 A3) = 0.02 (a) What is the probability that the system does not have a type 1 defect? (b) What is the probability that the system has both type 1 and type 2 defects? (c) What is the probability that the system has both type 1 and type 2 defects but not a type 3 defect? (d) What is the probability that the system has at most two of these defects?
Consider purchasing a system of audio components consisting of a receiver, a pair of speakers, and a CD player. Let A₁ be the event that the receiver functions properly throughout the warranty period, A₂ be the event that the speakers function properly throughout the warranty period, and A3 be the event that the CD player functions properly throughout the warranty period. Suppose that these events are (mutually) independent with P(A₁) = 0.92, P(A₂) = 0.96, and P(A3) = 0.90. (Round your answers to four decimal places.) (a) What is the probability that all three components function properly throughout the warranty period? (b) What is the probability that at least one component needs service during the warranty period? (c) What is the probability that all three components need service during the warranty period? (d) What is the probability that only the receiver needs service during the warranty period? (e) What is the probability that exactly one of the three components needs service…
A congressional committee is composed of 10 members: 6 Republicans and 4 Democrats. On any given vote, members vote independently. Whereas Republicans always vote their party line, Democrats vote their party line with probability .8, and vote Republican with probability .2. On a rainy day only 3 committee members (selected w/o replacement) were present. Let X = number of Democrats present Y = Number who voted Democratic that day Display the table of joint probabilities fX,Y(i,j) for 0 ≤ j ≤ i, for i=0,1,2,3. We can say that p(i,j)=P[x=i, Y=j].

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