Question 1206803
factory A produces 50% of all cars = .5
factory B produces 30% of all cars = .3
factory C produces 20% of all cars = .2


30% of cars produced in factory A are white = .3 * .5 = .15.
this means that 15% of all cars produced are white and produced in factory A.


40% of cars produced in factory B are white = .4 * .3 = .12.
this means that 12% of all cars produced are white and produced in factory B.


25% of cars produced in factory C are white = .25 * .2 = .05.
this means that 5% of all cars produced are white and produced in factory C.


the probability that the car is white is equal to .15 + .12 + .05 = .32.
this means that the probability that the car is white is 32%, regardless of which of the 3 factories produced it.


let D = the probability that the car is white = .32
let E = the probability that the car is produced in factory B = .3
let DE = the probability that the car is white and produced in factory B = .12


the probability that the car is produced in factory B given that the car is white is given by the equation p(E given D) = p(DE) / p(D) = .12 / .32 = .375.


to see how this works, take some fictitious number of cars produced and work it through, as i did in the following table that assumes 1000 cars were built in total.


50% of the cars were built in factory A = .5 * 1000 = 500
30% of the cars were built in factory B = .3 * 1000 = 300
20% of the cars were built in factory C = .2 * 1000 = 200


30% of the cars built in factory A were white = .3 * 500 = 150
40% of the cars built in factory B were white = .4 * 300 = 120
25% of the cars built in factory C were white = .25 * 200 = 50


total cars built were 1000
total white cars built were 320.
total cars built in factory B were 300.
total white cars built in factory B were 120.


number of all cars built in factory B divided by total number of all cars built = 300 / 1000 = .3.


number of all white cars built in factory B divided by total number of all cars built = 120 / 1000 = .12.


number of all white cars built in factory B divided by total number of all white cars built = 120 / 320 = .375.


your solutions are:


A. if an automobile produced by the company is selected at random, find the probability that is white.


that probability is .32 = 32%.



B. given that an automobile selected at random is white, find the probability that it came from factory B.


that probability is .375 = 37.5%.