100-year flood: Negative Binomial distribution

The negative Binomial distribution relates to independent trials and provides information on the probability of the number failures before  a certain number of successes. Continuing with our flood examples, the negative Binomial distribution can be used to determine the probability of the number of flood free years before a certain number of floods occur.

If Z is the number of flood free years, before r floods, and if a flood has a probability of occurrence of p in any year then:

$Z \sim nbinom(r, p)$

Example

A retirement village is vulnerable to the 100-year flood. If there are 3 or more floods in the next 20-years the political pressure will be such that the village will be relocated. What is the probability that this will occur?

Flooded retirement village (http://goo.gl/gtIdcz)

We need the probability that $Z + 3 \le 20$

The probability can be calculated using R as

pnbinom(17, 3, 0.01) = 0.001

So there isn’t much chance this will happen.

We can also calculate the probability using the Binomial distribution as 1 minus the probability of 2 or fewer flood in 20 years.

1 - pbinom(2, 20, 0.01) = 0.001

The expected value (mean) of Z, the number of flood free years before r floods, is:

$\frac{r(1-p)}{p}$

Example

What is the average number of years before the retirement village will experience 3 floods?

$\frac{3(1-0.01)}{0.01} = 297$

This is consistent with the average number of years between 1% floods being 100 years. On average we have 99 flood free years for each event, so in 300 years, on average we will have 297 flood free years and 3 flood events.