Question

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) 5 .95, P(A2) 5 .98, and P(A3) 5 .80.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 during the warranty period?f. What is the probability that all three components function properly throughout the warranty period but that at least one fails within a month after the warranty expires?

Answer 1

2.063X10^-4

0.999

0.833

3.26X10^-3

9.85x10^-3

Explanation:

B="At least one component needs service during the warranty period"

C="All three components need service during the warranty period"

D="Only the receiver needs service during the warranty period"

E="Exactly one of the three components needs service during the warranty period"

P(A1)=0.0595

P(A2)=0.0598

P(A3)=0.058

a) P(A1∩A2∩A3)=P(A1)P(A2)P(A3)=0.0595*0.0598*0.058=2.063X10^-4

b) P(B)=1-P(A1∩A2∩A3)=1-2.063X10^-4=0.999

c) P(C)=P(A1'∩A2'∩A3')=P(A1')P(A2')P(A3')=0.9405*0.9402*0.942=0.833

d) P(D)=P(A1'∩A2∩A3)=P(A1')P(A2)P(A3)=0.9405*0.0598*0.058=3.26X10^-3

e) P(E)=P(A1'∩A2∩A3)+P(A1∩A2'∩A3)+P(A1∩A2∩A3')=3.26X10^-3 + 0.0595*0.9402*0.058+0.0595*0.0598*0.942=9.85x10^-3