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(A,) = 0.16P(A, U A2) = 0.17P(A, U A3) = 0.13P(A,) = 0.09P(A, U A3) = 0.18P(A, N A, N Az) = 0.02P(A3) = 0.07(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)

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Author:Sheldon Ross

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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) = 0.16 P(A2) = 0.10 P(A3) = 0.07 P(A₁ UA2) = 0.18 P(A2 UA3) = 0.14 P(A₁ UA3) = 0.18 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?
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 chemical compound contains 3 particles of a reactant and 6 particles of a catalyst. Particles are removed from the compound at random, and after the removed particle is examined, it is returned back to the chemical compound together with an additional particle of the same type. We call this process of removal/return a "reaction". (a) Assume there are two consecutive reactions. Compute the probability that the first and the second removed particle is a reactant. (b) Compute the probability that at least one of the two removed particles is a reactant. (c) Assume now a third consecutive reaction occurs. Compute the probability that the third removed particle is a catalyst, given that exactly one of the first two is a reactant.
Suppose that 55% of all adults regularly consume coffee, 45% regularly consume carbonated soda, and 70% regularly consume at least one of these two products. (a) What is the probability that a randomly selected adult regularly consumes both coffee and soda? (b) What is the probability that a randomly selected adult doesn't regularly consume at least one of these two products?
Suppose that 50% of all adults regularly consume coffee, 40% regularly consume carbonated soda, and 65% regularly consume at least one of these two products. (a) What is the probability that a randomly selected adult regularly consumes both coffee and soda? (b) What is the probability that a randomly selected adult doesn't regularly consume at least one of these two products?
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?
A machine has four components, A, B, C, and D, set up in such a manner that all four parts must work for the machine to work properly. Assume the probability of one part working does not depend on the functionality of any of the other parts. Also assume that the probabilities of the individual parts working are P(A) = P(B) = 0.95, P(C) = 0.91, and P(D) = 0.96. Find the probability that at least one of the four parts will work. Round to six decimal places.
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(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?