In the task we have a 2 stage experiment:
- Selecting a device.
- Detecting if the chosen device is defective or not.
The probabilities in pt. 1 are as follows:
#D# - selecting a defective device #P(D)=0.16#
#N# - selecting a non-defective device #P(N)=0.84#
These numbers come from the sentence 'Of the devices checked 84% are not defective'
The second stage has the following events:
#DD# - detecting a defective device
#DN# - detecting a non defective device.
The probabilities depend on result of pt. 1:
If a defective device was chosen in 1. then:
#DD//D# - detecting a defective device as defective (correctly)
#DN//D# - detecting a defective device as non defective (incorectly)
#P(DD//D)=0.97#
#P(DN//D)=0.03#
If a non defective device was chosen in 1. then:
#DD//N# - detecting a non defective device as defective (incorectly)
#DN//N# - detecting a non defective device as non defective (corectly)
#P(DD//N)=0.09#
#P(DN//N)=0.91#
Now to calculate the probability of correct detection (#C#) we have to count:
#P(C)=P(D)*P(DD//D)+P(N)*P(DN//N)#
#P(C)=0.16*0.97+0.84*0.91=0.1552+0.7644=#
#=0.9196~~0.92#
#P(C)=0.92# or #P(C)=92%#
