Skip to main content

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.

Second blog in series here.
Third blog in series here.

Image: By Hovik Avetisyan [CC BY-SA 4.0 (http://creativecommons.org/licenses/by-sa/4.0)], via Wikimedia Commons

Comments

Popular posts from this blog

The Diamond Battery – your ideas for future energy generation

On Friday 25th November, at the Cabot Institute Annual Lecture, a new energy technology was unveiled that uses diamonds to generate electricity from nuclear waste. Researchers at the University of Bristol, led by Prof. Tom Scott, have created a prototype battery that incorporates radioactive Nickel-63 into a diamond, which is then able to generate a small electrical current.
Details of this technology can be found in our official press release here: http://www.bristol.ac.uk/news/2016/november/diamond-power.html.
Despite the low power of the batteries (relative to current technologies), they could have an exceptionally long lifespan, taking 5730 years to reach 50% battery power. Because of this, Professor Tom Scott explains:
“We envision these batteries to be used in situations where it is not feasible to charge or replace conventional batteries. Obvious applications would be in low-power electrical devices where long life of the energy source is needed, such as pacemakers, satellite…

Brexit: can research light the way?

What could Brexit mean for UK science? What impact will it have on UK fisheries? Could Brexit be bad news for emissions reductions? These were just some questions discussed at a Parliamentary conference last week, organised by the Parliamentary Office of Science and Technology (POST), the Commons Library and Parliament’s Universities Outreach team.

MPs researchers, Parliamentary staff and academic researchers from across the country came together to consider some of the key policy areas affected by the UK’s decision to leave the EU.

Why does academic research matter to Parliament? Given the unchartered waters that Parliament is facing as the UK prepares to withdraw from the EU, it is more important than ever that Parliamentary scrutiny and debate is informed by robust and reliable evidence.

Academic research is expected to meet rigorous standards of quality, independence and transparency. Although it is far from being the only source of evidence relevant to Parliament, it has vital ro…

A response to Trump's withdrawal from the Paris Agreement

The decision by President Trump to withdraw from the Paris Agreement on Climate Change puts the United States at odds with both science and global geopolitical norms.  The fundamentals of climate change remain unambiguous: greenhouse gas concentrations are increasing, they are increasing because of human action, the increase will cause warming, and that warming creates risks of extreme weather, food crises and sea level rise. That does not mean that scientists can predict all of the consequences of global warming, much work needs to be done, but the risks are both profound and clear. Nor do we know what the best solutions will be - there is need for a robust debate about the nature, fairness and efficacy of different decarbonisation policies and technologies as well as the balance of responsibility; the Paris Agreement, despite its faults with respect to obligation and enforcement, allowed great flexibility in that regard, which is why nearly every nation on Earth is a signatory.

Mor…