Question

72% of all vehicles examined at a certain emissions inspection station pass the inspection. Assuming that successive vehicles pass or fail independently of one another, calculate the probability that exactly one of the next three vehicles fail.

Answer 1

72% of all vehicles examined at a certain emissions inspection station pass the inspection.

p(pass) = 0.72 and p(fail) = 1 - 0.72 = 0.28

Let X be a number of vehicles fail among 3 vehicles.

Here, X ~ Binomial ( n = 3, p = 0.28)

probability mass function of X is,

P(X = x) = ⁿC_x p^x (1-p)^n-x

We want to find, P(X = 1)

P(X = 1) = ³C₁ * (0.28)¹ * (1 - 0.72)²

P(X = 1) = 3 * (0.28) * (0.72)²

P(X = 1) = 0.435456

Therefore, the probability that exactly one of the next three vehicles fail is 0.435456