Making More Sense of Probability
This article continues from Making Sense of Probability in Issue 118
One of the key issues with using probabilities in the real world is the way in which probabilities are stated as percentages and the way in which this information is then transmitted to the person needing to assess the probabilities, usually in some sort of assessment of risk (Gigerenzer 2003).
As an example, this might be a common statement of probabilities around a medical test for a disease:
‘On historical evidence, 0.8% of the population will develop a particular kind of cancer. When tested, 90% of those with cancer will get a positive result and 7% of those without cancer will get a positive result’.
For statisticians and medical practitioners who frequently work with such statements and fully understand them, the percentages are a clear assessment of both the risk of this type of cancer and the likely outcomes of the test. For the non-specialist this is probably unclear and using frequencies would probably be more helpful.
Compare the above with the following.
‘Out of 1000 people, on historical evidence, 8 are very likely to develop this type of cancer.
Of these 8 people, the test will successfully identify about 7 people and leave roughly one person with cancer undiagnosed.
Of the remaining 992 people, the test will be clear for 922 people but there are likely to be 70 people, who, as a result of the screening, get a positive test but do not have this type of cancer.’
or with this diagram.
Which do you think conveys the information in the simplest way to a non-specialist?
Now consider the following scenario.
‘In a court case, a witness sees a crime in their local town involving a taxi. This witness says the taxi was green. It is known from previous research by the police that witnesses are correct 80% of the time when making such statements. The police also know that 85% of taxis locally are blue and the other 15% are green. What is the probability that a green taxi was actually involved the crime?’
This scenario opens up a discussion around probabilities of combined events linked to the use of tree diagrams and the calculation of conditional probabilities.
Here, we want P (the taxi was green given that it was identified as green) which is the same as P (green and identified as green)/P(identified as green) = (0.15x0.8)/(0.15x0.8 + 0.85x0.2) = 0.41.
This is a lower probability than many jury members might expect and this would need careful explanation during the trial in relation to the low relative frequency of green taxis and the potentially poor quality of witness colour identification.
Are all your Core Maths/A Level Maths students using all the visual tools available to them in solving probability problems and could they all operate in a real world setting and explain probabilities to a non-specialist audience?
Gigerenzer, G. “Reckoning With Risk: Learning To Live With Uncertainty” (Penguin 2003)
Schneps and Colmez “Maths on Trial” (Basic Books 2013).
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