The mathematical language of probability has many uses. Arguably the most important is to quantify uncertainty about quantities, and answers to questions, of importance to decision-makers. A topical example might be the amount by which the global temperature of the planet earth will increase between now and 2050.

The natural view for a scientist to take is that quantification of uncertainty should be based on data and mathematical and statistical modelling of real-world phenomena based on scientific knowledge. The reality is that the data available are rarely, if ever, directly relevant to the decision-maker's question and that models are always incomplete descriptions. The consequence is that expert judgement is always needed as well. In many cases, expert judgement plays the primary role.

The talk will start with a discussion of how and why subjective probability is considered to be a good way to quantify uncertainty deriving from expert judgement and will then look at some of the standard tools for doing so using examples from the European Food Safety Authority's work. I will then look at the weaknesses of one of those tools and discuss a simple fix and some current work by a PhD student on ways to improve the fix. The mathematics involved is not very complex and is accessible to first year students. The hard part turns out to be finding theorems one can prove and which are also useful in the real world.