Question

A certain system can experience three different types of defects. Let A_i (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) = 0.11		P(A2) = 0.07		P(A3) = 0.05
P(A1 ∪ A2) = 0.15			P(A1 ∪ A3) = 0.14
P(A2 ∪ A3) = 0.1			P(A1 ∩ A2 ∩ A3) = 0.01

(Round your answers to two decimal places.)

(a) Given that the system has a type 1 defect, what is the probability that it has a type 2 defect?


(b) Given that the system has a type 1 defect, what is the probability that it has all three types of defects?


(c) Given that the system has at least one type of defect, what is the probability that it has exactly one type of defect?

Show all work and steps.

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Answer 1

from above":

P(A1)=			0.11
P(A2)=			0.07
P(A3)=			0.05
P(A1nA2)=P(A1)+P(A2)-P(A1 u A2)=			0.03
P(A2nA3)=P(A2)+P(A3)-P(A2 u A3)=			0.02
P(A1nA3)=P(A1)+P(A3)-P(A1 u A3)=			0.02
P(A1 U A2)=			0.15
P(A2 U A3)=			0.1
P(A1 U A3)=			0.14
P(A1nA2nA3)=P(A1)+P(A2)+P(A3)-P(A1nA2)-P(A1nA3)-P(A2nA3)+P(A1nA2nA3)=			0.01

a)

P(A2|A1) =P(A1 n A2)/P(A1)=

0.27

b)

P(A1nA2nA3|A1)=P(A1nA2nA3)/P(A1)=

0.09

c)

P(exactly one|at least one)=

0.12/0.17=

0.71