Question

f) The life of a power transmission tower is exponentially distributed, with mean life 25 years. If three towers, operated independently, are being erected at the same time, what is the probability that at least 2 will still stand after 35 years?

Answer 1

15.24% probability that at least 2 will still stand after 35 years

Explanation:

To solve this question, we need to understand the binomial distribution and the exponential distribution.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

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)`

The probability of finding a value higher than x is:

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

Probability of a single tower being standing after 35 years:

Single tower, so exponential.

Mean of 25 years, so
`m = 25, \mu = (1)/(25) = 0.04`

We have to find
`P(X > 35)`

`P(X > 35) = 1 - P(X \leq x) = 1 - (1 - e^(-\mu x)) = e^(-0.04*35) = 0.2466`

What is the probability that at least 2 will still stand after 35 years?

Now binomial.

Each tower has a 0.2466 probability of being standing after 35 years, so
`p = 0.2466`

3 towers, so
`n = 3`

We have to find:

`P(X \geq 2) = P(X = 2) + P(X = 3)`

In which

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 2) = C_(3,2).(0.2466)^(2).(0.7534)^(1) = 0.1374`

`P(X = 3) = C_(3,3).(0.2466)^(3).(0.7534)^(0) = 0.0150`

`P(X \geq 2) = P(X = 2) + P(X = 3) = 0.1374 + 0.0150 = 0.1524`

15.24% probability that at least 2 will still stand after 35 years