Five companies (A, B, C, D, and E) that make elec- trical relays compete each year to be the sole sup- plier of relays to a major automobile manufacturer. The auto company’s records show that the probabili- ties of choosing a company to be the sole supplier are Supplier chosen: A B C D E Probability: .20 .25 .15 .30 .10 a. Suppose that supplier E goes out of business this year, leaving the remaining four companies to compete with one another. What are the new probabilities of companies A, B, C, and D being chosen as the sole supplier this year? b. Suppose the auto company narrows the choice of suppliers to companies A and C. What is the probability that company A is chosen this year?

Answer:

a

  [tex]P(a | e') =  0.22[/tex]

  [tex]P(b | e') =  0.28[/tex]

  [tex]P(c | e') =  0.33[/tex]

b

  [tex]P(a | e' , d' , b') = 0.57  [/tex]

Step-by-step explanation:

From the question we are told that

   The probabilities are

Supplier  chosen            A                     B                    C            

Probability                P(a) = 0.20       P(b) =  0.25   P(c) =  0.15      

                                       D                      E

                                P(d) =  0.30     P(e) = 0.10

Generally the new probability of companies A being chosen as the sole supplier this year given that supplier E goes out of business is mathematically represented as below according to Bayes theorem

[tex]P(a | e') =  \frac{P (a \  and \  e')}{P(e')}[/tex]

      [tex]P(a | e') =  \frac{P (a)}{P(e')}[/tex]

     [tex]P(a | e') =  \frac{P (a)}{1- P(e)}[/tex]

=>   [tex]P(a | e') =  \frac{ 0.20}{1- 0.10}[/tex]

=>   [tex]P(a | e') =  0.22[/tex]

Generally the new probability of companies B  being chosen as the sole supplier this year given that supplier E goes out of business is mathematically represented as below according to Bayes theorem

[tex]P(b | e') =  \frac{P (b \  and \  e')}{P(e')}[/tex]

      [tex]P(b | e') =  \frac{P (b)}{P(e')}[/tex]

     [tex]P(b | e') =  \frac{P (b)}{1- P(e)}[/tex]

=>   [tex]P(b | e') =  \frac{ 0.25}{1- 0.10}[/tex]

=>   [tex]P(b | e') =  0.28[/tex]

Generally the new probability of companies C  being chosen as the sole supplier this year given that supplier E goes out of business is mathematically represented as below according to Bayes theorem

[tex]P(c | e') =  \frac{P (c \  and \  e')}{P(e')}[/tex]

      [tex]P(c | e') =  \frac{P (c)}{P(e')}[/tex]

     [tex]P(c | e') =  \frac{P (c)}{1- P(e)}[/tex]

=>   [tex]P(c | e') =  \frac{ 0.15}{1- 0.10}[/tex]

=>   [tex]P(c | e') =  0.17[/tex]

Generally the new probability of companies D  being chosen as the sole supplier this year given that supplier E goes out of business is mathematically represented as below according to Bayes theorem

[tex]P(d | e') =  \frac{P (d \  and \  e')}{P(e')}[/tex]

      [tex]P(d | e') =  \frac{P (d)}{P(e')}[/tex]

     [tex]P(d | e') =  \frac{P (d)}{1- P(e)}[/tex]

=>   [tex]P(d | e') =  \frac{ 0.30}{1- 0.10}[/tex]

=>   [tex]P(c | e') =  0.33[/tex]

Generally the probability that  B, D , E  are not chosen this year is mathematically represented as

      [tex]P(N) =  1 - [P(e) +P(b) + P(d) ][/tex]

=>       [tex]P(N) =  1 - [0.10 +0.25  +0.30 ][/tex]

=>       [tex]P(N) =  0.35[/tex]

Generally the probability that A is chosen given that E , D , B  are rejected this year is mathematically represented  as

      [tex]P(a | e' , d' , b') =  \frac{P(a)}{P(N)}[/tex]

=>     [tex]P(a | e' , d' , b') =  \frac{0.20 }{0.35 }[/tex]    

=>     [tex]P(a | e' , d' , b') = 0.57  [/tex]

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