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:

**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?

We need the probability that

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:

**Example**

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

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.

**Further reading**

Jones, O, Maillardet, R. and Robinson, A. (2009) Scientific programming and simulation using R. CRC Press

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