A machine will be used to produce up to 5 units a day. After each in completed,

Question

A machine will be used to produce up to 5 units a day. After each in completed, there is a 20% probability that the machine becomes inoperable for the day (it will be definitely repaired before the next morning). If less than 3 units are produced in a day, then the production level is considered unsatisfactory and additional measures will be required. i. Find the probability of an unsatisfactory performance in one day. ii. Suppose that the additional measures cost $1000 per day, and by introducing new practices you could reduce the probability of failure after each unit to 10%. How much would you be willing to pay for this change? Assume that you gain no profit other than reduction in costs. iii. Without changes described in (b), how much would you allocate for costs related to unsat- isfactory performance in an annual budget. Justify your answer. Can you guarantee that the budget will not be exceeded? A machine will be used to produce up to 5 units a day. After each in completed, there is a 20% probability that the machine becomes inoperable for the day (it will be definitely repaired before the next morning). If less than 3 units are produced in a day, then the production level is considered unsatisfactory and additional measures will be required. i. Find the probability of an unsatisfactory performance in one day. ii. Suppose that the additional measures cost $1000 per day, and by introducing new practices you could reduce the probability of failure after each unit to 10%. How much would you be willing to pay for this change? Assume that you gain no profit other than reduction in costs. iii. Without changes described in (b), how much would you allocate for costs related to unsat- isfactory performance in an annual budget. Justify your answer. Can you guarantee that the budget will not be exceeded? A machine will be used to produce up to 5 units a day. After each in completed, there is a 20% probability that the machine becomes inoperable for the day (it will be definitely repaired before the next morning). If less than 3 units are produced in a day, then the production level is considered unsatisfactory and additional measures will be required. i. Find the probability of an unsatisfactory performance in one day. ii. Suppose that the additional measures cost $1000 per day, and by introducing new practices you could reduce the probability of failure after each unit to 10%. How much would you be willing to pay for this change? Assume that you gain no profit other than reduction in costs. iii. Without changes described in (b), how much would you allocate for costs related to unsat- isfactory performance in an annual budget. Justify your answer. Can you guarantee that the budget will not be exceeded?

Explanation / Answer

Answer to part i)

P(x<3) = P(x=0) + P(x=1) + P(x=2)

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Using Binomial probability formula”

P(X=x) = nCx * p^x * (1-p)^(n-x)

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Given:

N = 5

P = 0.20

.

P(x=0) = 5C0 * 0.20^0 * 0.80^5 = 0.3277

P(x=1) = 5C1 * 0.20^1 * 0.80^4 = 0.4096

P(x=2) = 5C2 * 0.20^2 * 0.80^3 = 0.2048

P(x<3) = 0.3277 +0.4096 +0.2048

P(x<3) = 0.9421

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Answer to part ii)

P(x<3) for 10% probability is:

P(x=0) = 5C0 * 0.10^0 * 0.90^5 = 0.5905

P(x=1) = 5C1 * 0.10^1 * 0.90^4 = 0.3281

P(x=2) = 5C2 * 0.10^2 * 0.90^3 = 0.0729

P(x<3) = 0.5905 + 0.3281 + 0.0729

P(x<3) = 0.99915

1000 * 0.99915 = 999.15

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Answer to part iii)

1000*0.9421 = 942.1

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