Question

3. A car manufacturer offers a 5yr/Unlimited km warranty on one of its models. Historically, 8% of cars sold will require repairs under the terms of this warranty. Suppose a dealer sells 190 cars of that model and let X be the number of those cars that will require service under the terms of the warranty. (a) A suitable probability model for X is a binomial distribution. What are the parameters of the binomial model in this case. Justify the use of a binomial model in context. (b) Find the probability that exactly 16 cars require service under the warranty. (c) Find the probability that at least 16 cars require service under the warranty. 2 2

Answer 1

the terms of (a) There Service or not. Since of cars X denotes the number of cars that will require Service under the warranty. two possibilities. Either the car requires X denotes the number requiring service, let p be the probability of cars that will require repairing. Then dendes the probability of cars that won't require repairing. Since there are two possible outcomes; Probabilities, and a we have it q = 1 X follows a Binomial distribution Sample size, n = 190 80% of cars require repairing: Hence, p= 0.8 with q Hence

n- 190_2 X Then q = 0.2 (.p+q=1) :: X follows a Binomial distribution will parameters; n=190, = 0.8, 9=0.2 :: Pdf, p(x) = nCx px q ; 2=0,1, ---, PCX) 190 Cx (0.8) (0.2);a=0,1,...,190 (8) PC exactly 16 cars =P[x=16] = 190 C 16 (0.2) 190 C 16 (0.8)! (0.27th 190-16 require service) (0.816 COOP [at least 16 cars require service =P[x216] -1- P[x216] [PCX70) + P(x+1)+ --. +p[x = 15)) - [190 Co (0.8)*(0.2)". 1906, (0.8) (0.2) + 190615 Co.8)'% (0.27775] 189 +