Answer:
P(A|B) = P(A∩B)/P(B) = 100%
Which means that there is 100% probability that the item has at least one type of defect given that the item has only a shaft defect.
Step-by-step explanation:
Conditional probability P(A|B) can be expressed as;
P(A|B) = P(A∩B)/P(B) .....1
Given;
30% will have a shaft defect,
15% will have a bushing defect,
and 65% will be defect-free
Total probability = 100% = P(shaft or/and bushing defect) + P(defect free)
P(shaft or/and bushing defect) = 100% - P(defect free)
= 100% - 65% = 35%
And
P(shaft or/and bushing defect) = P(shaft def only) + P(bushing def only) + P(shaft and bushing defect)
P(shaft or/and bushing defect) = P(shaft defect) + P(bushing defect) - P(shaft and bushing defect)
Substituting the values we have;
35% = 30% + 15% - P(shaft and bushing defect)
P(shaft and bushing defect) = 45% - 35% = 10%
Let A="The item has at least one type of defect"; and B="The item has only a shaft defect".
P(A) = P(shaft or/and bushing defect) = 35%
P(B) = P(shaft only defect) = 30% - 10% = 20%
P(A∩B) = 20%
Substituting into equation 1
P(A|B) = P(A∩B)/P(B) = 20%/20%
P(A|B) = 1/1 = 100%
Which means that there is 100% probability that the item has at least one type of defect given that the item has only a shaft defect.
