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4. In a new industrial facility, accidents occur infrequently. It is known that the probability of...

4. In a new industrial facility, accidents occur infrequently. It is known that the probability of an accident on any given day is 0.05 and accidents are independent of each other. If this new industrial facility begins its function today, [5+5+5+5=20 points] (a) what is the expected number of days needed to encounter the first accident in this industrial facility? (b) what is the probability that the first accident occurs on the the day after tomorrow? (c) what is the conditional probability that the first accident occurs no earlier than on the 5th day given that the first 2 days are accident-free? (d) what is the conditional probability that the first accident occurs no earlier than on the 23rd day given that the first 20 days are accident-free?

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Answer #1

a)expected number of days needed to encounter the first accident in this industrial facility=1/0.05 =20

b)

probability that the first accident occurs on the the day after tomorrow =P(not today, not tomorrow & on day after tomorrow) =0.952 *0.05 =0.0451

c)

conditional probability that the first accident occurs no earlier than on the 5th day given that the first 2 days are accident-free

=P(no accident in next 2 days)=0.952 =0.9025

d)

conditional probability that the first accident occurs no earlier than on the 23rd day given that the first 20 days are accident-free =P(no accident in next 2 days)=0.952 =0.9025

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