Question

A maintenance firm has gathered the following information regarding the failure mechanisms for air conditioningsystems:

Yes No
Evidence of electrical failure Yes 55 17
Electrical failure No 32 3


The units without evidence of gas leaks or electrical failure showed other types of failure. If this is a representative sample of AC failure, find the probability.

a. That failure involves a gas leak
b. That there is evidence of electrical failure given that there was a gas leak
c. That there is evidence of a gas leak given that there is evidence of electrical failure

Answer 1

A) 0.8131 ≈ 0.81

B) 0.6322 ≈ 0.63

C) 0.7639 ≈ 0.77

Explanation:

A)

Total number of units

= 55 + 32 + 17 + 3

= 107

hence the probability that the failure involves a gas leak

= ( 55 + 32 ) / 107

= 0.8131

B) probability that there is evidence of electrical failure given tat there is a gas leak

P ( A / B ) =
`(P(A n B))/(P(B))`

P( A∩B ) = 55/107 = 0.5140

P ( B ) = ( 55 + 32 ) / 107 = 0.8131

hence P ( A / B ) = 0.5140 / 0.8131 = 0.6322

A = failure due to electrical

B = failure due to gas leak

C) Probability of a gas leak given that there is evidence of electrical failure

P ( B / A ) =
`(P(BnA))/(P(A))`

P ( B ∩ A ) = 0.5140

P( A ) = ( 55 + 17 ) / 107 = 0.6729

hence P ( B / A ) = 0.5140 / 0.6729 = 0.7639