It seems, to me at least, that probability is one of ‘those topics’ that students find to be more difficult than others. Each year I have winced slightly as it looms on the horizon of the scheme of work and each year I’ve varied my approach to try and get them (and keep them) on board. This post focusses on conditional probability, now designated as an A-level (not AS) topic.
Students should already have a fair amount of experience with tree diagrams and Venn diagrams. They should also have met the terms independent and mutually exclusive – I think that’s actually where the problems begin. Why are these concepts so challenging? Perhaps because one (independent) is a term borrowed from every day language, but in that standard usage it tends to mean ‘separate from’ which is actually closer to the meaning of mutually exclusive. Secondly, because we try to give students an intuitive and descriptive feeling for what independent means in statistics, they then lose sight of the fact that it has a precise, mathematical definition.
What would I do about this? If I could, I would separate the teaching of the two terms with as much time as possible. We could teach ‘mutually exclusive’ when we cover sampling techniques (the strata for a stratified sample should be mutually exclusive, for example). More contentiously, I wonder if we shouldn’t introduce ‘independent’ until the time we study conditional probability? Compare:
Definition 1: A and B are independent events if
Definition 2: A and B are independent events if
These are equivalent*, but I would argue the second gives a much better feeling for what statistical independence really means.
When teaching conditional probability this year, I demonstrated it separately in three contexts: two-way tables, Venn diagrams and tree diagrams. Prior to the discussion of conditional probability, of course I wanted to check that students brought their prior understanding to the surface – through a mixture of starter questions on the board (two-way tables; tree diagrams), or some mini-whiteboard work (Venn diagrams).
In each context I adopted the same approach: a probability question involving the typical ‘given that’ phrasing; a highlighted, restricted part of the diagram; and the formula for conditional probability. Each time, I described the highlighter approach as intuitive and the formula approach as ‘safer’ – not susceptible to misinterpretation. Each time, we also considered the reverse of the conditional statement, ie. we contrasted and .
I split these three contexts over three separate lessons. This gave the students the opportunity to focus on each one fully, in isolation, and to also experience the recall of the formula for conditional probability each day. The examples below aren’t the exact ones that I used in my lessons, but should be illustrative enough.
In a two-way table
In a Venn diagram
To find , we can think intuitively:
In a tree diagram
Intuitively: just read from the relevant branch.
Note: there there is a significant opportunity for confusion here: the calculation for is the product , however this is unrelated to the product formula for independent events. Indeed, the events ‘able to construct’ and ‘pass’ are not independent here.
Results and Looking Ahead
We set weekly class tests in our department and thus get fairly rapid feedback about how well students have picked up new topics. I was pleased to see that my emphasis of using the slightly more abstract (but secure) approach given by the formula for conditional probability made an appearance in almost all of the students’ work. I was a little disheartened (but not completely surprised) by the proportion of incorrect answers to the very first question on the test:
1 (a) State what it means for events A and B to be independent.
(b) State what it means for events A and B to be mutually exclusive.
We still have 6 or 7 months to sort that out!
I’ll be returning to conditional probability at every available opportunity. Due to the ordering of our scheme of work, I am now moving on to discrete random variables and thus can include conditional questions there. This is also followed by the binomial distribution (which many of you may well have already taught in the AS year). I find it somewhat surprising that conditional probability has been designed to come after hypothesis testing as, within a hypothesis test, the probability we calculate is conditional on H0 being true. With the order of our own scheme of work, I can emphasise that conditional statement more heavily and hopefully the students will gain a better appreciation for the process we follow in this type of hypothesis test.
The definitions are not quite equivalent… The reason the first is the preferred definition for independent events is that the second cannot be used in the case where . In all other cases, they are equivalent!