Question

The average lifetime of a set of tires is three years. The manufacturer will replace any set of tires failing within two years of the date of purchase. The lifetime of these tires is known to follow an exponential distribution. What is the probability that the tires will fail within two years of the date of purchase?

Answer 1

48.67% probability that the tires will fail within two years of the date of purchase

Explanation:

Exponential distribution:

The exponential probability distribution, with mean m, is described by the following equation:

`f(x) = \mu e^(-\mu x)`

In which
`\mu = (1)/(m)` is the decay parameter.

The probability that x is lower or equal to a is given by:

`P(X \leq x) = \int\limits^a_0 {f(x)} \, dx`

Which has the following solution:

`P(X \leq x) = 1 - e^(-\mu x)`

In this question:

`m = 3, \mu = \frac{1}[3}`

`P(X \leq 2) = 1 - e^{-(2)/(3)} = 0.4867`

48.67% probability that the tires will fail within two years of the date of purchase