Please explain by calculation, thank you!

A manufacturing process has a 3% defect rate. Inspectors catch 95% of defects but also fail 5% of nondefective parts. If we pick a part at random from all those that pass inspection, what is the probability that part is actually defective? Is the event "inspection outcome correct" independent of the outcome of the manufacturing process?

a)5%
b)0.16%
c)0.5%

correct option is b)0.16%

Step-by-step explanation:

given data

defect rate P(defect ) = 3 % = 0.03

Inspectors catch = 95%  = 0.95

fail nondefective =  5% = 0.05

solution

we get here  P(No Defect)  that is

P(No Defect)  = 1 - P( Defect)   .........1

P(No Defect)  = 1 - 0.03

P(No Defect)  = 0.97

and

P(Pass | Defect)  will be

P(Pass | Defect)  = 1 - Inspectors catch   ...........2

P(Pass | Defect) = 1 - 0.95

P(Pass | Defect)  = 0.05  

and

P(Pass | No Defect) is

P(Pass | No Defect) = 1 - fail non defective    ..............3

P(Pass | No Defect) = 1 - 0.05

P(Pass | No Defect) =  0.95

so now we apply here Baye's theorem

so

P(Defective | Pass)  is

P(Defective | Pass)  =  

P(Defective | Pass)  = 0.0016

P(Defective | Pass)  = 0.16 %

so correct option is b)0.16%

1/3

step-by-step explanation:

to find this, you can divide 12 by 4 which is 3 or you can multiply 4 until you reach 12 (4x3=12). both should give you 3, meaning theres three times less bananas than there are oranges.