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A new automated production process averages 1.5 breakdowns per day. Because of the cost associated with a breakdown, management is concerned about the possibility of having three or more breakdowns during a day. Assume that breakdowns occur randomly, that the probability of a breakdown is the same for any two time intervals of equal length, and that breakdowns in one period are independent of breakdowns in other periods. What is the probability of having three or more breakdowns during a day?

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STATISTICS F/BUSINESS+ECONOMICS-TE...

13th Edition
Anderson
Publisher: CENGAGE L
ISBN: 9781305881884

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BuyFindarrow_forward

STATISTICS F/BUSINESS+ECONOMICS-TE...

13th Edition
Anderson
Publisher: CENGAGE L
ISBN: 9781305881884
Chapter 5, Problem 70SE
Textbook Problem
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A new automated production process averages 1.5 breakdowns per day. Because of the cost associated with a breakdown, management is concerned about the possibility of having three or more breakdowns during a day. Assume that breakdowns occur randomly, that the probability of a breakdown is the same for any two time intervals of equal length, and that breakdowns in one period are independent of breakdowns in other periods. What is the probability of having three or more breakdowns during a day?

To determine

Compute the probability of having 3 or more breakdowns during a day.

Explanation of Solution

Calculation:

There is an average of 1.5 break downs per day for a new automated production process. The breakdowns occur randomly and the probability of breakdown is the same for any two time intervals of equal length. The break down in one period is independent of the breakdowns in other periods.

Here, the random variable x defined as the number of break downs per day for a new automated production process.

Here, the occurrence of break downs is random and the break down in one period is independent of the breakdowns in other periods. The mean number of break downs per day is 1.5. That is, the occurrences (break downs) are uniformly distributed for every day. Thus, the number of break downs per day for a new automated production process has Poisson distribution with parameter, μ=1.5.

The Poisson probability function is,

f(x)=μxeμx!

Where, x is the number occurrences of the event in a given interval, μ is the mean occurrences of the event, and e is a constant that has value 2.71828.

The probability that 3 or more breakdowns during a day is obtained as shown below:

P(3 or more)=P(x3)=1P(x2)=1[f(0)+f(1)+f(2)]

The required probability is obtained by substituting the values μ as 1.5 and the corresponding x values in the Poisson probability function. That is,

P(3 or more)=1[f(0)+f(1)+f(2)]=1[(1

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