Question

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 1

Answer:

a

  $P(a | e') = 0.22$

  $P(b | e') = 0.28$

  $P(c | e') = 0.33$

b

  $P(a | e' , d' , b') = 0.57$

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

$P(a | e') = \frac{P (a \ and \ e')}{P(e')}$

      $P(a | e') = \frac{P (a)}{P(e')}$

     $P(a | e') = \frac{P (a)}{1- P(e)}$

=>   $P(a | e') = \frac{ 0.20}{1- 0.10}$

=>   $P(a | e') = 0.22$

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

$P(b | e') = \frac{P (b \ and \ e')}{P(e')}$

      $P(b | e') = \frac{P (b)}{P(e')}$

     $P(b | e') = \frac{P (b)}{1- P(e)}$

=>   $P(b | e') = \frac{ 0.25}{1- 0.10}$

=>   $P(b | e') = 0.28$

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

$P(c | e') = \frac{P (c \ and \ e')}{P(e')}$

      $P(c | e') = \frac{P (c)}{P(e')}$

     $P(c | e') = \frac{P (c)}{1- P(e)}$

=>   $P(c | e') = \frac{ 0.15}{1- 0.10}$

=>   $P(c | e') = 0.17$

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

$P(d | e') = \frac{P (d \ and \ e')}{P(e')}$

      $P(d | e') = \frac{P (d)}{P(e')}$

     $P(d | e') = \frac{P (d)}{1- P(e)}$

=>   $P(d | e') = \frac{ 0.30}{1- 0.10}$

=>   $P(c | e') = 0.33$

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

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

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

=>       $P(N) = 0.35$

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

      $P(a | e' , d' , b') = \frac{P(a)}{P(N)}$

=>     $P(a | e' , d' , b') = \frac{0.20 }{0.35 }$    

=>     $P(a | e' , d' , b') = 0.57$