Question

All Seasons Plumbing has two service trucks that frequently need repair. If the probability the first truck is available is .73, the probability the second truck is available is .59, and the probability that both trucks are available is .43: What is the probability neither truck is available

Answer 1

Answer: .11

Explanation:

Let F be the event that the first truck is available and S be the event that the second truck is available.

The probability of neither truck being available is expressed as P(
`F^(C)`∩
`S^(C)`) , where P(
`F^(C)`) is the probability that the event F doesn't happen and P(
`S^(C)`) is the probability that the event S doesn't happen.

P(
`F^(C)`)= 1-P(F) = 1-0.73 = 0.27

P(
`S^(C)`)=1-P(S) = 1-0.59 = 0.41

Since
`F^(C)` and
`S^(C)` aren't mutually exclusive events, then:

P(
`F^(C)`∪
`S^(C)`) = P(
`F^(C)`) + P(
`S^(C)`) - P(
`F^(C)`∩
`S^(C)`)

Isolating the probability that interests us:

P(
`F^(C)`∩
`S^(C)`)= P(
`F^(C)`) + P(
`S^(C)`)- P(
`F^(C)`∪
`S^(C)`)

Where P(
`F^(C)`∪
`S^(C)`) = 1 - 0.43 = 0.57

Finally:

P(
`F^(C)`∩
`S^(C)`) = 0.27+ 0.41 - 0.57 = 0.11