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:

a

  [tex]P(A) =  0.85[/tex]

b

  [tex]P(A | D)  =  0.7522 [/tex]

Step-by-step 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  

  [tex]P(A) =  0.85[/tex]

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

        [tex]P(A | D)  =  \frac{P(A) *  p}{P(D)}[/tex]

Here

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

       [tex]P(D) =  P(A) *  p  +  P(B) *  q  +  P(C) *  r[/tex]

=>     [tex]P(D) =  085 *  0.025   +  0.10  *  0.040  +  0.05  *  0.06[/tex]

=>     [tex]P(D) =  0.02825 [/tex]

So

      [tex]P(A | D)  =  \frac{ 0.85 * 0.025 }{0.02825}[/tex]

=>    [tex]P(A | D)  =  0.7522 [/tex]

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