Question

The waiting time for a fire department to get called to a house fire is exponentially distributed with an average wait time of 14 minutes. Given that it has already taken 11 minutes, what is the probability that the wait time will be more than an additional 16 minutes?
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Answer 1

Answer:

0.319 = 31.9% probability that the wait time will be more than an additional 16 minutes

Step-by-step explanation:

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

Exponential distribution:

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

$f(x) = \mu e^{-\mu x}$

In which $\mu = \frac{1}{m}$ 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:

$P(X > x) = 1 - P(X \leq x) = 1 - (1 - e^{-\mu x}) = e^{-\mu x}$

Conditional Probability

We use the conditional probability formula to solve this question. It is

$P(B|A) = \frac{P(A \cap B)}{P(A)}$

In which

P(B|A) is the probability of event B happening, given that A happened.

$P(A \cap B)$ is the probability of both A and B happening.

P(A) is the probability of A happening.

In this question:

Event A: It has already taken 11 minutes.

Event B: It will take 16 more minutes.

Exponentially distributed with an average wait time of 14 minutes.

This means that $m = 14, \mu = \frac{1}{14}$

Probability of the waiting time being of at least 11 minutes:

$P(A) = P(X > 11) = e^{-\frac{11}{14}} = 0.4558$

Probability of the waiting time being of at least 11 minutes, and more than an additional 16 minutes:

More than 11 + 16 = 27 minutes. So

$P(A \cap B) = P(X > 27) = e^{-\frac{27}{14}} = 0.1454$

What is the probability that the wait time will be more than an additional 16 minutes?

$P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.1454}{0.4558} = 0.319$

0.319 = 31.9% probability that the wait time will be more than an additional 16 minutes