Question

The occurrence of unplanned shutdown in a manufacturing industry can be considered as a Poisson process. On average, eight shutdown occurs per year.

(a) Calculate the probability that

• the first shutdown occurs during the first quarter;

• more than one shutdown occurs during the first half of the year.

(b) Calculate the conditional probability that the second shutdown will occur during the first half of the year if the first shutdown occurred during the first quarter of the year.

Answer 1

Answer:-

Given That:-

The occurrence of unplanned shutdown in a manufacturing industry can be considered as a Poisson process. On average, eight shutdown occurs per year.

Given,

shutdown occurs per year

(a) Calculate the probability that

• the first shutdown occurs during the first quarter

Expected number of shutdown in first quarter = 8/4

= 2

P(First shutdown occur in first quarter) = 1 - P(No shutdown in fisrt quarter)

  

= 1 - 0.1353

= 0.8647

P(X = 0) = 0.8647

• more than one shutdown occurs during the first half of the year.

P(more than 1 shutdown in first half) =

= 1 - P(X = 0) - P(X = 1)

Expected number of shutdown in first half () = 8/2

= 4

= 1 - P(X = 0) - P(X = 1)

= 1 - [0.0183 + 0.0733]

= 1- 0.0916

(b) Calculate the conditional probability that the second shutdown will occur during the first half of the year if the first shutdown occurred during the first quarter of the year.

Here If X is the number of shutdown time occurs in half year then

Expected number of downtime on each half = 4

As we know that past events doesn't impact the future probabilities. In case of poission Distribution So, whatever happened in first shutdown doesn't impact the second shutdown.

P(Second shutdown occurs in second quarter first shutdown occur in first quarter) = P(shutdown occur in second quarter)

= 1 - P(No shutdown in first quarter)

= 1 - 0.0183

= 0.9817

So, the required Probability is = 0.9817

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