What are the major misconceptions that students have when it comes to combining events in probability? Specifically, when the question does not explicitly ask for a tree diagram, what strategies to students employ, and what mistakes do they make?

Using real life data and explanations from students all around the world from my Diagnostic Questions website, we can find out! :-)

Have a look at the question below.

What percentage of students do you think get it correct?
What is the most popular incorrect answer?
What explanations do students give for the incorrect answers?

Give the question a go yourself, and try to come up with an explanation of the correct answer that would make sense to students:

At the time of writing, this question has been answered 111 times, and has been answered correctly just 33% of the time. By far the most popular incorrect answer is A, which 48% of students go for,

To see this data, read student explanations, and filter by things such as gender and age, just visit the Data Question page.

As we all know, probability can be a minefield of misconceptions. Students are rarely sure when to add and when to multiply, and even when they can remember the rules, they rarely understand why they are doing what they are doing.

Indeed, when faced with the traditional tree diagrams probability question in the higher tier GCSE exam, many students are absolutely fine. They go into the well rehearsed routine of filling out the branches, and then multiplying the numbers going across and adding them to go down. So long as they are careful, they will help themselves to the 4 or 5 marks on offer without breaking sweat or over-exercising their brain.

But it is questions like the one above that cause problems, because there is no mention of a tree diagram. The question calls for a deeper level of understanding from the students, and each of the wrong answers teases out a key, and all too common, misconception.

Let’s see what expatiations they students gave, in their own words:

Correct Explanations for A

There are a couple of beautiful student explanations here for what is a pretty tricky concept to explain:

If you think about a tree diagram, there is a 1/4 change of getting a 2 on the first spinner, and a 1/4 chance on the second spinner. If you need both to happen, you must multiply, which gives you 1/16

There are 16 different scores you could get (4 x 4). Only one of them gives you two lots of 2. So the answers must be 1/16

Interestingly, a few students get this question correct, but via their explanations we can see that their understanding is not quite right:

i think this because there is only 1 chance of getting a 2 and there are 16 numbers altogether

I think this is the answer because there is the number 1 on each spinner and if you put the two one’s together you get 1/8

This illustrates the importance of reading students’ explanations, and not just assuming because they have got it right that they understand.

Incorrect Explanations for A

A was by far the most popular choice for students. And their explanations reveal exactly the misconception that is all too common with probability:

There are 4 bits on each spinner which make the spinners up. 4 add 4 equals to 8 and there is one 2’s on each spinner which adds up too 2. So therefore we have 2 chances in 8 of getting a two on both spinners.

i think this because there are two 2’s and there are 8 numbers altogether this means that it is 2 over 8

you have a 1/4 chance of getting a 2 on each dice and then you double that

There are 8 numbers in total 2 of them are 2

As a teacher, how would you deal with that misconception?

Incorrect Explanations for B

B tempted in a few students. And I would argue it is probably the best wrong answer, as the idea of 16 outcomes is shining through. But where does the 2 come from?

i think this is the answer because there are 2 sets of 2s and there are 16 sections

Incorrect Explanations for D

Not many go for D, but like B there is some understanding evident here. The 1 in the numerator suggests that students have correctly identified there is only one possible outcome (as opposed to two) that gives what we are looking for:

there is one 2 on each spinner, so you need to multiply these, and 1 x 1 = 1. And there are 8 different outcomes (1 to 4 and 1 to 4), so it must be 1/8

So, how do we deal with all of these? Do we resort to a tree diagram? Will that give us the understanding we need, or just fall back on the same old rules? How about a sample space diagram showing all the possible outcomes? Any other suggestions?

This question is taken from the Probability with Spinners Quiz, which provides more challenges like this one:

Below is a video showing you how to access all the pages and information discussed above: