The lifetime of a lightbulb can be modeled with an exponential random variable with an expected lifetime of 1000 days. (a) Find the probability that the lightbulb will function for more than 2000 days. (b) Find the probability that the lightbulb will function for more than 2000 days, given that it is still functional after 500 days.

a) 13.53% probability that the lightbulb will function for more than 2000 days.

b) 22.31% probability that the lightbulb will function for more than 2000 days, given that it is still functional after 500 days.

Step-by-step explanation:

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

In which is the decay parameter.

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

Which has the following solution:

In this problem, we have that:

So

(a) Find the probability that the lightbulb will function for more than 2000 days.

Either it will function for 2000 days or less, or it will function for more than 2000 days. The sum of the probabilities of these events is decimal 1. So

We want P(X > 2000). So

In which

13.53% probability that the lightbulb will function for more than 2000 days.

(b) Find the probability that the lightbulb will function for more than 2000 days, given that it is still functional after 500 days.

This is:

From a, we have that P(X > 2000) = 0.1353.

We also have that, following the same logic as a):

In which

So

22.31% probability that the lightbulb will function for more than 2000 days, given that it is still functional after 500 days.