Question

There are three companies that produce a critical electronic navigation component (ENC) used in the aerospace industry. These companies are Alice Manufacturing, Byte International, and Cognizant Technologies. Alice makes 85% of the ENCs, Byte makes 10%, and Cognizant makes the remaining 5%. The ENCs made by Alice have a 2.5% rate of defects, the ones made by Byte have a 4.0% rate of defects, and the ones made by Cognizant have a 6% rate of defects. If an ENC is randomly selected from the general population of ENCs, the probability that it was made by Alice is _______. If this ENC is later tested and found to be defective, the probability that it was made by Alice is _______.

Answer 1

a

`P(A) =  0.85`

b

`P(A | D)  =  0.7522`

Explanation:

From the question we are told that

The percentage of ENC made by Alice is P(A) 85% = 0.85

The percentage of ENC made by Byte P(B) 10%=0.10

The percentage of ENC made by Cognizant P(C) = 5%= 0.05

The percentage rate of defect of ENC made by Alice is p = 2.5 % = 0.025

The percentage rate of defect of ENC made by Byte is q = 4.0% = 0.04

The percentage rate of defect of ENC made by Cognizant r = 6% =0.06

Generally the probability that a randomly selected ENC will be made by Alice is

`P(A) =  0.85`

Generally the probability that a randomly selected ENC that was found to be defective will be made by Alice is

`P(A | D)  =  (P(A) *  p)/(P(D))`

Here

`P(D)` is the probability that a randomly selected ENC will be defective and this is mathematically represented as

`P(D) =  P(A) *  p  +  P(B) *  q  +  P(C) *  r`

=>
`P(D) =  085 *  0.025   +  0.10  *  0.040  +  0.05  *  0.06`

=>
`P(D) =  0.02825`

So

`P(A | D)  =  ( 0.85 * 0.025 )/(0.02825)`

=>
`P(A | D)  =  0.7522`