Question

I'm having trouble applying bayes formula with the following multi-part question

In April 2013, the total sales from General Motors, Ford, or Chrysler was 606,334 cars or light trucks. The probability that the vehicle sold was made by General Motors was 0.392, by Ford 0.350, by Chrysler 0.258. Additionally, the probability that a General Motors vehicle sold was a car was 0.395, a Ford vehicle sold was a car was 0.370, and a Chrysler vehicle sold was a car was 0.332.

(1) Given the vehicle sold was a car, find the probability it was made by General Motors

(a) About 0.332 ; (b) About 0.274 ; (c) About 0.376 ; (d) About 0.232 ; (e) About 0.418 ;

(2) Given the vehicle sold was a car, find the probability it was made by Chrysler.

(a) About 0.376 ; (b) About 0.232 ; (c) About 0.332 ; (d) About 0.274 ; (e) About 0.418 ;

(3) Given the vehicle sold was a light truck, find the probability it was made by General Motors.

(a) About 0.418 ; (b) About 0.232 ; (c) About 0.376 ; (d) About 0.274 ; (e) About 0.332 ;

(4) Given the vehicle sold was a light truck, find the probability it was made by Chrysler.

(a) About 0.274 ; (b) About 0.332 ; (c) About 0.418 ; (d) About 0.232 ; (e) About 0.376 ;

Answer 1

We are given here that:
P( GM ) = 0.392,
P( Ford ) = 0.35, and
P( Chrysler ) = 0.258

Also, we are given here that:
P( car | GM) = 0.395,
P( car | Ford ) = 0.370, and
P( car | Chrysler ) = 0.332

1) Using law of total probability, we have here:
P( car ) = P( car | GM)P( GM ) + P( car | Ford)P( Ford ) + P( car | Chrysler )P( Chrysler )
P( car ) = 0.392*0.395 + 0.35*0.37 + 0.258*0.332 = 0.369996

Using bayes theorem, we have here:
P( GM | car ) = P( car | GM)P( GM ) / P(car ) = 0.392*0.395 / 0.369996 = 0.4185

Therefore e) 0.418 is the required probability here.

2) Again, using bayes theorem, we get here:

P( Chrysler | car ) = P( car | Chrysler)P( Chrysler ) / P(car ) = 0.258*0.332 / 0.369996 = 0.2315

Therefore b) 0.232 is the required probability here.

3) We know here that: P( light truck ) = 1 - P(car ) = 1 - 0.369996 = 0.630004
P( light truck | GM) = 1 - P(car | GM) = 1 - 0.395 = 0.605

Therefore, using bayes theorem we get here:
P( GM | light truck ) = P(light truck | GM)P(GM) / P( light truck ) = 0.605*0.392 /  0.630004 = 0.3764

Therefore c) 0.376 is the required probability here.

4) P(light truck | Chrysler) = 1 - P( car | Chrysler ) = 1 - 0.332 = 0.668

Therefore, using bayes theorem we get here:
P( Chrysler | light truck ) = P(light truck | Chrysler)P(Chrysler) / P( light truck ) = 0.668*0.258 /  0.630004 = 0.2735

Therefore a) 0.274 is the required probability here.