A car manufacturing company makes a particular model of car in three versions: t

Question

A car manufacturing company makes a particular model of car in three versions: two-door, four-door and hatchback. The proportion of the different versions of car that are made are 60%, 30% and 10% respectively. Cars with two-doors have a GPS device with probability 0.2, and an automatic transmission with probability 0.3, and have both a GPS and automatic transmission with probability 0.1. 15% of cars have four-doors and a GPS, while 5% are four-door with automatic transmission. None of the four-door cars have both a GPS and automatic transmission. All hatchbacks have both GPS and automatic transmission. The sample space is = {all cars}. We can formulate events as follows: T = “two door”; F = “four door”; H = “hatchback”; G = “GPS”; A = “automatic transmission”.

From the information given, find the probability that a randomly selected car from this company has a GPS.

From the information given, find the probability that randomly selected two-door car from this company has a GPS or automatic transmission.

Find the probability that a randomly selected four-door car from this company has automatic transmission.

Mr. Smith buys a car from this car company. First he notices that it has a GPS. What is the probability that the car has two doors?

Explanation / Answer

Pr(T) = 0.60 ; Pr(F) = 0.30 ; Pr(H) = 0.10

Pr(G l T) = 0.2 ; Pr(A l T) = 0.3 Pr(G A l T) = 0.1

Pr(F G) = 0.15 ; Pr(F A) = 0.05

Pr( G A l H) = 1

Question 1 From the information given, find the probability that a randomly selected car from this company has a GPS.

Answer : Pr(GPS) = P(T) * Pr(G l T) + P(F) * Pr(G l F) + P(H) * P(G l H) = 0.60 * 0.2 + 0.15 + 0.1 = 0.37

QUestion 2 From the information given, find the probability that randomly selected two-door car from this company has a GPS or automatic transmission.

Answer : Pr(GPS or automatic transmission) = Pr(G U A l T) * Pr(T) + Pr(G U A l F) * Pr(F) + Pr(G U A l H) * Pr(H)

(Pr(G U A) = (0.2 + 0.3 -0.1) * 0.6 + 0.15 + 0.05 + 1 * 0.1 = 0.54

Question 3 Find the probability that a randomly selected four-door car from this company has automatic transmission.

Answer : Pr(A l F) = Pr(A F)/ Pr(F) = 0.05/0.30 = 1/6 = 0.1667

Question 4 Mr. Smith buys a car from this car company. First he notices that it has a GPS. What is the probability that the car has two doors?

Answer : Pr(T l G) = Pr(G l T) * Pr(T)/ [P(T) * Pr(G l T) + P(F) * Pr(G l F) + P(H) * P(G l H)]

= ( 0.6 * 0.2)/ (0.6 * 0.2 + 0.15 + 0.1) = 0.3243

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