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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4. We have an M/M/2 queueing system. The arrival rate is 2 per hour if there are fewer than 2 customers in the system and 3 otherwise. The service rate is 3 customers per hour per server. (a) (b) Find the limiting probabilities Find the expected number of customers in the system.
An assembly plant receives 50% of its tubes from supplier X, 30% from supplier Y and 20% from supplier Z. 2% of supplier X's tubes, 4% of supplier Y's tubes and 9% of supplier Z's tubes are defective. A tube is selected at random from the assembly plant. (i) Find the probability that the tube is not defective. If the tube is defective, find the probability that it is supplied by Z. (ii) (iii) Find the probability that the tube is defective or it is supplied by X.
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.
Suppose A and B are two events with probabilities: Р (4°) 3 40, Р(В) %— .25, Р(A U B) %3D75. Find the following: a) P(AN B). b) Р(A). с) Р(В°). d) P((AN B)°).
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.
A device can fail in four different ways with probabilities T1= 0.5, T2 = 0.1, T3 = 0.1, and 14 = 0.3. Suppose there are 10 devices that fail independently of one another. What is the probability of 3 failures of the first kind, 2 of the second, 1 of the third, and 4 of the fourth?
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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)?
Suppose, in an airport, there are three runways. In the ideal situation, the probabilities, 2/9, 1/6, and 11/18 represent the likelihood that individual runways 1, 2 and3 respectively are accessed by a randomly arriving commercial jet. If 6 airplanes are arriving at the airport then what is the probability that there are 2, 1, and 3 airplanes are accessed by runway 1, runway 2, and runway respectively.
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…

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