### CFA Practice Question

Blue Line Truck Rentals and Cramer Trucks compete in the personal moving market. Both Blue Line and Cramer have white and blue trucks. Blue Line's fleet consists of 80% blue trucks and the rest white trucks. Cramer's fleet has 60% white trucks and 40% blue trucks. Blue Line has 5,000 trucks whereas Cramer has 3,000 trucks. If a white truck is spotted on the road, what is the chance that it belongs to Blue Line?
A. 20%
B. 35.7%
C. 64.3%
Explanation: Blue Line has 5/8th of the trucks on the road and Cramer has 3/8th of the trucks on the road. Thus, the fraction of trucks that are white and belong to Blue Line = (5/8) x 0.2. The fraction of trucks that are white and belong to Cramer = (3/8) x 0.6. If a white truck is spotted, the probability that it belongs to Blue Line: Prob(Blue Line | White truck) = Prob(Blue Line & White truck) / Prob(White truck) = [(5/8) x 0.2] / [(5/8) x 0.2 + (3/8) x 0.6] = 0.357, or 35.7%

User Comment
bikegeek I think it is easier, faster, and more logical to find out number of white trucks Blue Line has and divide by the total number of white trucks on road. IS this the wrong way to approach these problems?
achu I solved for the number of trucks- thought that was the prferred way, actually- and got this right.
thekapila bikegeek its right though..
surob good question
mattg Prob(A given B)= Prob(AB)/ Prob(B)

For this problem, Prob (BluLine given Wht) = Prob(BluLn+Wht)/Pro(Wht truck)
maria15 I agree with bikegeek
Sam123456 You can solve it via bikegeek's method if you have the number of trucks in each fleet or the number of trucks in total and the number of trucks in one of the fleets. Now you know how many trucks of each colour are in each fleet.

But if instead you are told that 62.5% (or 5/8) of trucks are Blue Line and they don't tell you any number of trucks, then you would have to use the formula.
aniketcpp Using Bayes' theorem it was very simple and took me 10 sec only. :)
bergje11 Using Bayes':
P(Blueline|white truck) = P(white truck|Blueline)*P(Blueline) / P(white truck)
P(white truck|Blueline) = 20% given
P(Blueline) = 5000 / (5000+3000) = 62.5%
P(white truck) = (20%*5000 + 60%*3000)/8000
P(Blueline|white truck) = 20%*62.5% / 35%
ashish100 This took me so long. Hope this helps someone who's confused.

P(a) = 5/8 i.e. P(BlueLineBus)
P(b)= 2800/8000 i.e. P(WhiteBus)
P(ab) IS NOT P(a) * P(b)!!

P(BlueLineBus that is White) = 5/8 * .20 or P(ab) = .1250

Now use the P(ab) = P(a|b) * P(B) formula.
mlaique Though I understand that Bayes theorem is the ideal way to solve, another way is to write out a grid:

B Trucks I C Trucks
White 20% I 60%
Blue 80% I 40%
Total Trucks 5000 I 3000

B Trucks that are white: 20% * 5000 = 1000

Total white trucks = 20%*5000 + 60%*3000 = 2800

Therefore:

1000/(2800) = 35.7%