In my experience, students often find it difficult to discern between the concepts of union, intersection, independence, conditional probability, mutually exclusive events, and so on. They often attempt to apply a set of formulae, without a sense of where they come from.
This is a brief account of a lesson introducing probability theorems to year 12 students. It was based on these images sent to me by David Butler (@DavidKButlerUoA), a maths teacher from Australia, who had made them to help a student visualise (statistical) independence.
I decided to use these images to create a set of questions that might allow students to come to an awareness of probabilistic relationships for themselves. Here are the questions I created:
In part 2, I wanted students to gain fluency in working with intersection, union and conditional probability. In part 3 I hoped that students might realise that some of the formulae shown are identities, whilst others only hold in special cases.
I recognise these questions are a bit 'lusty', and with more thought I might have found a more subtle way of bring students to these awarenesses. For example: the second student extract below shows a student recognising the interplay between P(AnB) and P(AuB), before doing part 3.
In part 3 I wanted students to become aware that (b) and (d) are identities - well, true for all of these diagrams at least - whilst the other formulae were only sometimes true. These 'special' ones (middle and bottom right) are then starting points for discussion about independent and mutually exclusive events.
Here is some of the work that students produced over the hour:
Many students identified a connection between union and intersection, and others (working in fractions) noticed a connection between intersection and conditional probabilities:
I like the fact students are using words to make their own conclusions, reflecting while doing:
In the next example, we can see a student systematically identifying which of the proposed identities seem always true, and which are only sometimes true:
Tomorrow, we are going to review the work and bring together these ideas. I will draw attention to the central image and the last image.
By comparing students' findings, we may see that the following are only true for the central image:
P(A|B) = P(A)
P(B|A) = P(B)
Similarly, the following is only true for the final image:
P(AuB) = P(A) + P(B)
P(AnB) = 0
Hopefully we will discover that these are special cases of the two identities, which we may then go on to prove:
P(AnB) = P(A).P(B|A)
P(AuB) = P(A) + P(B) - P(AnB)
*Having read this, David has suggested that an interesting follow-up might be to give empty grids and invite students to make diagrams with given properties, a lovely example of doing and undoing.