Aircraft Defects Hydraulic landing assemblies coming from an

Aircraft Defects. Hydraulic landing assemblies coming from an aircraft rework facility inspected for defects. Historical records indicate that 8% have defects in shafts only, 6% have defects in bushings only, and 16% have at least one of the two types of defects. One of the assemblies is selected at random. What is the probability that it: a) Has both types of defects? b) Has a bushing defect? c) Has exactly one of the two types of defects? d) Has neither type of defect?

Solution

Let event s represent a Shaft defect and event b represent a Brushing defect.
The given probabilities are:
P(s ¬b) = 0.08
P(b ¬s) = 0.06
P(b s) = 0.02
Note: these are mutually exclusive

What you wish to find are:
a: Brushing defect.
P(b) = P(b ¬s) + P(b s)
  => .06+0.02= 0.08

b: Shaft or Brushing.
P(s U b) = P(b ¬s) + P(s ¬b) + P(b s)
=> 0.06+0.08+0.02 = 0.16
c: Shaft and not Brushing OR Brushing and not Shaft
P((s ¬b) U (b ¬s)) = P(s ¬b) + P(b ¬s)
= 0.06+0.08 = 0.14
d: Not Shaft and not Brushing
P(¬s ¬b) = 1 - P(s U b) = 1-0.116 = 0.84