Probability

Question

The probability that a given amplifier malfunctions at a given gig is 5%.  A musician has two amplifiers on tour.  At the beginning of the fourth gig, the first amplifier malfunctions; it is pushed to the side, and the back up amplifier is plugged in.  What is the probability that it works OK for this gig?

Answer 1

One way to look at it is that there is nothing about the behavior of the first amplified that affects anything about the second amplifier.  So the probability that this amplifier fails in this one gig is still the same 5% that was originally stated.  So the probability that it will work OK is 95%.

 

This can also be seen as a problem of conditional probability.  They are asking, what is the probability that the second amp will fail given that the first one has already failed.  The formula for this is P (A|B) = P (A∩B) / P(B).

 

But the failure of the two amps are independent events.  So P (A∩B) = P(A) * P(B).  So P (A∩B) / P(B) = P(A) * P(B) / P(B) = P(A).  So the probability that the second amp will fail given that the first has already failed is just the independent probability that the second amp will fail, which is 5%.  So the probability that it will work OK is again 95%.