A system consists of two components. The probability that the second component functions in a satisfactory...

A system consists of two components. The probability that the second component functions in a satisfactory manner during its design life is 0.9, the probability that at least one of the two components does so is 0.94, and the probability that both components do so is 0.85. Given that the first component functions in a satisfactory manner throughout its design life, what is the probability that the second one does also? (Round your answer to three decimal places.)

Solutions

Expert Solution

Let A be the event that the first component functions in a satisfactory manner during its design life.

Let B be the event that the second component functions in a satisfactory manner during its design life.

The probability that the second component functions in a satisfactory manner during its design life is 0.9. Therefore, P(B) = 0.9

The probability that at least one of the two components functions in a satisfactory manner during its design life is 0.94. Therefore, P(A B) = 0.94

The probability that both components functions in a satisfactory manner during its design life is 0.85. Therefore, P(A B) = 0.85

Given that the first component functions in a satisfactory manner throughout its design life, the probability that the second one does also is

P(B | A) = P(A B) / P(A) ..........(conditional probability)

Using the addition rule,

P(A B) = P(A) + P(B) - P(A B)

0.94 = P(A) + 0.9 - 0.85

P(A) = 0.94 - 0.9 + 0.85

P(A) = 0.89

Now, P(B | A) = 0.85 / 0.89

= 0.955

Therefore, given that the first component functions in a satisfactory manner throughout its design life, the probability that the second one does also is 0.955

orchestra answered 1 year ago

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