Exam Question: Weather

The Royal Society for Statistics (RSS) has been critical of the teaching of probability, and in particular a British exam question on cycling, and this one:

If the probability of there being a drop in temperature tomorrow is0.8 and the probability of it
raining is 0.5, what is the probability of neither event happening

What do you make of it? Please consider before reading on.

Do you agree with their assessment, here:?

Mathematics has never been well taught to the majority of English school-children. My parents complained about their inability at mathematics, and many of my contemporaries boasted about their lack of mathematical knowledge. There is no easy solution but one of the contemporary methods, making mathematics appear relevant, fails dismally in the teaching of probability, as the following examples from national examination papers demonstrate.

What might be wrong with this first example? Please consider before reading on.

The implied assumption of independence in these examples is both mathematically naive and scientifically wrong. It undermines good teaching in both areas and instead of being relevant is misguided and misleading.

What do you make of this? Any other issues? Please consider before reading on.

I have commented on the cycling question, showing how a broader interpretation of the mathematics of uncertainty can be applied, much of which can also be applied here, so here I comment more on the specifics of the question, which help illustrate the general points previously made.

The RSS make the following suggestion:

Teaching probability through the old-fashioned medium of balls in urns would at least give our citizens an appreciation of their chance of winning the lottery, as well as the potential to understand that scientific models of uncertainty are analogues of these simple systems.

But how would this help? You might like to consider this before considering my speculations. (I am not an educationalist.)

The clause ‘the probability of there being a drop in temperature tomorrow is 0.8’ suggests an analogy with drawing balls out of an urn with 10 balls, 8 of which are labelled ‘drop’. The next clause suggest drawing balls out of an urn with 10 balls, 5 of which are labelled rain. Hence one might think the probability required is the probability of drawing two blanks balls, yielding what is presumed to be the answer that the exam board considered ‘correct’.

The RSS comment about independence reflects a different analogy: that there is a single urn of many balls, that may be labelled ‘no event’, ‘drop’, ‘rain’ or ‘drop and rain’. 80% are labelled ‘drop’ or ‘drop and rain’, 50% ‘rain’ or ‘drop and rain’. But we are not told how many are labelled ‘drop and rain’, and so (as in my comments on the previous question) can only say that number of balls labelled ‘no event’ must be between 0% and 20%. We cannot tell what the precise value is, thus we cannot determine a precise probability without neither extra information or invoking some statistical principle, such as assuming independence by default. But in this case, drops in temperature and rain are both associated with cold fronts, and so the default would be nonsense.

The most one could say would be ‘at most 20%, if you want a better answer ask a meteorologist’. Even then, if the given probabilities are derived from past weather data and the required probability concerns the future (e.g., later today) then one should be concerned about climate change.

See Also

Other puzzles.

Dave Marsay

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