5  The probabilistic representation of extremes

Extreme events are not predictable in the medium or long term. That raises the need for a probabilistic representation, in which the probability of occurrence of these events is estimated. Extreme rainfall is an example of this: annual probabilities of the occurrence of rainfall of various intensities are estimated. This fact sheet explains how these probabilities are calculated and sets out the main underlying assumptions.

Consider rainfall events with a 5% probability of occurring each year. The return period, which corresponds to the average number of years separating two consecutive occurrences of this event, is T = 1/0.05 = 20 years. A rarer event with a return period of 100 years would have a 1% probability of occurring each year.

This probabilistic representation is completely equivalent to a lottery. For rainfall with a 5% probability of occurrence, or 1/20, the likelihood of the occurrence of such an event in a given year is equivalent to rolling a twenty-sided die (Figure 5.1). If a particular number—say the number 20—is rolled, then the event in question occurs; otherwise, it does not. This implies that each outcome is statistically independent of the previous outcome, i.e. the probability of rolling a 20 on each roll of the die remains 1/20, regardless of the outcome of the previous roll1. So, it’s possible to roll a 20 five times in a row, even though the probability is very low (less than one chance in 3 million). Table 5.1 gives the probabilities of rolling a given number 0, 1, 2, 3, 4, or 5 or more times when a 20-sided die is rolled 20 times. We can see that the most likely number of times is 1, with a probability of 37.7%, but that it is very likely that the number will not be rolled (35.8%) or that it will be rolled more than once (26.5%)2.

Table 5.1: Probabilities of rolling a given number 0, 1, 2, 3, 4, or 5 or more times when a 20-sided die is rolled 20 times.
Number of times the number is rolled Probability
0 35.8%
1 37.7%
2 18.9%
3 6.0%
4 1.3%
5 or more 0.3%
Figure 5.1: A 20-sided die.

  1. Statistical independence in the case of extreme rainfall remains a hypothesis, since certain periodic phenomena such as El Niño can cause the annual maximum rainfall recorded during an El Niño cycle to be more or less correlated, depending on the region considered.↩︎

  2. The mean number of occurrences of an event of probability \(p\) after \(N\) draws is \(p N\).↩︎