### Converting probabilities between time-intervals

This is the first in an irregular sequence of snippets about some of the slightly more technical aspects of uncertainty and risk assessment.  If you have a slightly more technical question, then please email me and I will try to answer it with a snippet.

Suppose that an event has a probability of 0.015 (or 1.5%) of happening at least once in the next five years. Then the probability of the event happening at least once in the next year is 0.015 / 5 = 0.003 (or 0.3%), and the probability of it happening at least once in the next 20 years is 0.015 * 4 = 0.06 (or 6%).

Here is the rule for scaling probabilities to different time intervals: if both probabilities (the original one and the new one) are no larger than 0.1 (or 10%), then simply multiply the original probability by the ratio of the new time-interval to the original time-interval, to find the new probability.

This rule is an approximation which breaks down if either of the probabilities is greater than 0.1. For example, to scale a probability of 0.04 in the next 5 years up to 20 years we cannot simply multiply by 4, because the result, 0.16 (or 16%), is larger than 0.1. In this case we have to use the proper rule, which is

p_new = 1 - (1 - p_orig)^(int_new / int_orig)

where ‘^’ reads ‘to the power of’. The example above becomes

p_new = 1 - (1 - 0.04)^(20 / 5) = 0.15 (or 15%).

So the approximation would have been 1 percentage point out in this case. The highlighted text in yellow can be pasted directly into a spreadsheet cell (the answer is 0.1507).

Of course it is unlikely to matter in practice whether the probability is 0.15 or 0.16.  But the difference gets bigger as the probabilities get bigger.  For example, it would definitely be a mistake to multiply a 0.25 one-year probability by 5 to find the five-year probability, because the result would be greater than 1.  Using the formula, the correct answer is a five-year probability of 0.76.

Blog post by Prof. Jonathan Rougier, Professor of Statistical Science.