Question

The lifetime of a leaf blower is exponentially distributed with a mean of 5 years. If you buy a 12 year old leaf blower, what is the probability that it will be working after an additional 5 years?

Answer 1

Answer:

1.967 × 10⁻¹¹

Step-by-step explanation:

This is a conditional probability problem

Probability that a 12 year old leaf blower works for 5 more years = Probabilty that a leaf blower works for 17 years, given that it has worked for 12 years = P(17|12)

P(17|12) = P(17 n 12)/P(12)

P(17 n 12) = P(17)

Exponential random variable is given as

f(x) = λ e^(-λ.x)

λ = rate parameter = 5 years per blower

x = variable whose probability is required = 17

f(17) = 17 e^(-5×17)

f(17) = 17 e⁻⁸⁵ = 2.067 × 10⁻³⁶

f(12) = 12 e^(-5×12)

f(12) = 12 e⁻⁶⁰ = 1.051 × 10⁻²⁵

P(17|12) = (2.067 × 10⁻³⁶)/(1.051 × 10⁻²⁵)

P(17|12) = 1.967 × 10⁻¹¹