Question

An engineer is concerned about the potential loss due to failures of equipment. He obtained the following information regarding the failure events:

There are two possible failure types F1 and F2, which are unrelated;

F1 occurs with 3% in 10 years probability;

F2 occurs with 7% in 15 years probability, and

Associated costs in case of failures for F1 and F2 are $1000 and $400, respectively.

(a) Which failure event (F1 or F2) is more likely to happen during a year of operation?

(b) What is the expected loss due to the failures F1 and F2 in a year?

(c) What is the probability of at least one failure in 5 years?

Answer 1

SOLUTION:

From given data,

An engineer is concerned about the potential loss due to failures of equipment. He obtained the following information regarding the failure events:

There are two possible failure types F1 and F2, which are unrelated;

F1 occurs with 3% in 10 years probability;

F2 occurs with 7% in 15 years probability, and

Associated costs in case of failures for F1 and F2 are $1000 and $400, respectively.

F1 occurs with 3% in 10 years probability;

Therefore , each year F1 occurs with 3/10 = 0.3% probability

F2 occurs with 7% in 15 years probability.

Therefore , each year F2 occurs with 7/15 = 0.4667% probability.

(a) Which failure event (F1 or F2) is more likely to happen during a year of operation?

Failure event F2 is more likely to happen during a year of operation.

(b) What is the expected loss due to the failures F1 and F2 in a year?

The expected loss due to the failures F1 and F2 in a year is

= 1000* (0.3/100) + 400*(0.4667/100)

= 3+ 1.8668

= $ 4.8668

The expected loss due to the failures F1 and F2 in a year is $ 4.8668

(c) What is the probability of at least one failure in 5 years?

P(at least one failure in 5 years) = 1 - P(no failure in 5 years)

P(at least one failure in 5 years) = 1 - (0.997*0.995)⁵

P(at least one failure in 5 years) = 0.039

Therfore,

3.9% is the probability of at least one failure in 5 years