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:

The following question is taken from my website Diagnostic Questions. Here you will find 1000s of high quality maths questions, ideal for assessment for learning, which have been created and shared by maths teachers all over the world.

I love this question. Tree Diagrams are a very popular question on GCSE exams, they carry a significant number of marks, most students find them pretty straight forward, and yet there are countless opportunities to make some very costly mistakes that result from misconceptions students might have.

This question helps me as a teacher identify those misconceptions my students may have very quickly. Therefore, it helps me decide if I need to spend more time going over Tree Diagrams -and exactly which aspects I need to focus on - without having to spend 15 minutes waiting for students to draw one from scratch.

Having asked the question, let's see what we might learn from our students' responses:

Answer a)
might suggest they have attempted to multiply the two probabilities together, but have a misconception when it comes to multiplying two decimals together.

Answer b)
is a classic. This is where students have decided to add the two probabilities instead of multiplying them.

Answer c)
is an interesting one. Students selecting this answer may believe that as there are 4 possible outcomes, each one is equally likely, and hence the probability of each is 0.25.

Answer d)
is the correct answer

In the 5 minutes it took me to ask my Year 11s this question, them to vote with their fingers for their choice of answer, and then listen to them make the cases for each of their answers, I learnt so much about the areas I needed to focus on during that lesson - multiplying decimals, as it turned out! The students also learnt loads from listening to each other's explanations and justifications. Furthermore, even though no-one selected Answer d), it led to a really interesting discussion about why it was one of the options.

A really lovely question, and thanks so much to JCMurphy for uploading it

To access the question to see statistics for each answer, to include it in a quiz, download it for offline use, embed it in a blog, or loads more, just click here.And if you have used the question, or have any thoughts or comments about it - perhaps you would include a different incorrect answer - then please share your comments below. And please consider creating a question yourself on the website!