# Exam Question: Cycling

The following British exam question is quoted by the Royal Society for Statistics (RSS).

John cycles to school on average 3 days out of 5. Bill cycles to school on average 2 days out of 5. Find the probability that on a certain day they will both cycle to school.

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 agree with their criticism, but I have four other issues with this type of question:

1. Its the kind of question that one gets in school exams, not real life.
2. The question implies an approach that is deeply unscientific.
3. In so far as the real issue is about ‘probability’, there are better ways to tackle it than the (numeric) Bayesian approach that the RSS commend.
4. Even if the answer were a ‘probability’, it ought to be made clear what sort of probability and how to interpret it.
5. It seems to me that framing this type of problem in terms of ‘probability’ brings along all sorts of controversial baggage that is best avoided.
6. The way the question is posed is highly suggestive, but while it may seem reasonable to assume that the questioner has good grounds for their implicit assumptions, maybe we should sometimes question them?

To expand:

1. If you can observe when the two pupils cycle, why not take note of when they cycle together, and use the observed frequency as the estimate of ‘probability’?
2. If anyone thinks they have a way of solving the problem from the data provided, shouldn’t they test it out? (In which case, wouldn’t they have abandoned it long ago – if only for the reasons the RSS give.)
3. From Boole’s point of view, for example, it could be that John and Bill choose to cycle together when they can, so that they both cycle to school on two days out of 5. Alternatively, perhaps they job share (e.g., they are brothers who care for an ill mum). In this case, they may never cycle together. Intermediate cases are possible. It gets worse. If one were to conjecture about John and Bill’s relationship prior to knowing how often they cycle to school, a lawyer might argue that the fact that 2+3=5 is ‘evidence’ in support of the conjecture that they job share, whereas cycling equally often would have supported the conjecture that they cycled together.
4. From a mathematical perspective, it is clear how to calculate with whatever probabilities are appropriate, but there are different kinds of probabilities and different ways of resolving ambiguities, and it is a scientific question, not a mathematical one, as to how to interpret probabilities in any particular case. In this case, I was taught to preface my answers with ‘making the usual assumptions ..’ while muttering under my breath ‘this is nonesense, but it’s what I need to do to get the marks’.
5. In this case one could take the frequency data as given (which does not involve probabilities) and deduce the possible frequency range for them both cycling, as in (3) above, leaving it to others to convert to whatever kind of ‘probability’ suits them. More helpfully, one could join with subject matter experts in more fully characterising circumstances in which cycling together may be more or less frequent.
6. Finally, saying that ‘John cycles to school on average …’ seems to suggest that this average is meaningful. And ‘Find the probability that on a certain day they will both cycle to school’ seems to suggest that we can consider a day in the future (or why bother?). But maybe winter is coming on? Maybe his bike is not so reliable? Etc.

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.

Again, you might like to consider this before considering my speculations. (I am not an educationalist.)

Urn problems involve the notion of random sampling, which has nothing obvious to do with the given question. One way to interpret this type of application is that the mathematics tells you what to expect if the sampling is random. There is no mathematical test for randomness (e.g. most computers can generate ‘random’ numbers that aren’t actually random, but it doesn’t matter.) So if any application of probability theory appears to violate ‘the mathematics of uncertainty’, a mathematician can simply claim that the sampling can’t have been truly random. That is, the best one can say is that some situation appears random, and any ‘prediction’ will be conditional on the situation remaining random with an unchanged distribution. (E.g., on the weather not changing or the bicycle not failing.