probability Q2 A study was carried out to investigate the causes of power failur

Question

probability Q2

A study was carried out to investigate the causes of power failures, and yield the following results 10% were due to transformer damage, 75% and 5% were due to line problem, and both the transformer and line problems respectively. Calculate the probabilities of the power failures caused by (a) transformer damage given that there was line problem (b) line problem but not the transformer damage. (c) transformer damage or line problem. Determine, by giving reasons, whether the above two causes are independent or mutually exclusive.

Explanation / Answer

From information given, P(power failure due to transformer damage)=0.10, P(power failure due to line problem)=0.75, and P(power failure due to transformer and line problem)=0.05

a. P(power failure due to transformer damage| line problem)=P(power failure due to line problem and transformer damage)/P(line problem)=0.05/0.75 [using rule for conditional probability, P(B|A)=P(A and B)/P(A)]

=0.0667 (ans)

b. P(line problem and transformer damage')=0.75*(1-0.10) [using general multiplication rule, P(A and B)=P(A)*P(B), P(B')=1-P(B)]

=0.675 (ans)

c. P(transformer damage or line problem)=P(power failure due to transformer damage)+P(power failure due to line problem)=0.10+0.75 [using general addition rule, P(A or B)=P(A)+P(B)]

=0.85 (ans)

Declare two events independent, if P(power failure due to transfomer damage|line damage)=P(power failure due to transformer problem)

Here,  P(power failure due to transformer damage| line problem)=0.0667 and P(power failure due to transformer damage)=0.10

Because, 0.6667=/=0.10, the events are not independent.

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