# Probability Tree Diagrams: Maths Diagnostic Question of the Week 22

What are the major misconceptions that students have when it comes to probability tree diagrams? Specifically, care students able to correctly interpret the meaning of the question and select which branches they require?

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 433 times, and has been answered correctly 52% of the time. The most popular inocrrect answers given are C with 21% and B with 19%.

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

Probability tree diagrams are a staple of the maths GCSE. Students usually encounter one for the first time around Year 8, and keep bumping into them every school year thereafter.

When you think about it, there is a lot that could go wrong with a tree diagram question. Firstly, there is the construction of one from scratch if it is not given in the question. Secondly, there is the (potentially huge) issue of being able to multiple and add the fractions or decimals that may be involved in them.

But on top of all that, there is the concept of the tree diagram itself. Which, when you come to think about it, is rather bizarre. In the example above, students are essentially being asked to perceive four alternate realities. The first of which entails a win in the first game, followed by a win in the second game. Then, in a parallel universe, we ask them to consider what would happen if they won the first game but then went on to lose the second game. And so on. And for some students, this can be a little much to get their heads around.

This is why I like this particular question. The tree diagram has already been constructed, the branches are labelled, and the answers have been chosen in such a way as to avoid any confusion with the decimal operations themselves. Hence, we can focus all our attention on the students' understanding of the concept of a tree diagram, and specifically can they interpret what is meant by winning "at least one game".

Let's have a look at how the students themselves interpret this:

Correct Explanations for A

Many students explained the correct answer beautifully:

I could win the first and win the second. 0.6 x 0.6 = 0.36. I could win the first and lose the second 0.6 x 0.4 = 0.24. I could also lose the first and win the second 0.4 x 0.6 = 0.24 0.36+0.24+0.24 = 0.84

Looking at the tree diagram you can see that there are four outcomes (win + win, win + lose, lose + win, lose + lose). Three of these give at least one win. The probability of a win and a win is 0.36 (0.6 x 0.6 you times because its an 'and'- win And win- not an 'or'). The probability of a win and a lose is 0.24 (0.6 x 0.4). The probability of a lose and then a win is 0.24 (0.4 x 0.6). Then add up these three probabilities (add because its an 'or'- you want the probability of a win + win OR a win + lose OR a lose + win). 0.36 + 0.24 + 0.24= 0.84

At least one win= win and win or win and lose or lose and win. When its an 'and' you times (e.g. win And win) so the probabilities of each of these are: win and win=0.6 x 0.6=0.36, win and lose=0.6 x 0.4= 0.24, lose and win= 0.4 x 0.6= 0.24. Then when its an 'or' (e.g. win+win Or win+lose) you add so the probability of at least one win is 0.36+0.24+0.24= 0.84.

Go along the first branch and there are two possible scenarios of winning at least once. So 0.6 x 0.6 = 0,36 + (0.6x0.4) is 0.6 Then on the bottom branch you can only win once by 0.6x0.4 which is 0.24. Add that on to 0.6 and you get 0.84

A textbook could not have written it better. And all it takes is for one of these explanations to make sense to your students, and their misconception is well on its way to being resolved.

Incorrect Explanations for B

B attracted almost 1/5th of responses, and you may be able to guess why. This is not down to an inability to multiply decimals, it is doe to a misconception with the concept of following and understanding the tree diagram itself.

it could be win then loss which would be 0.4*0.6, or loss then win which would be 0.6*0.4, both of which would be 0.24.

Because a win then a lose has the probability of 0.4/1 and a lose then a win has the probability of 0.6/1 and those multiplied together=0.24/1=0.24

I think this is the answer because You times togethor 0.6 and 0.4 to get 0.24

follow the path of win then loss then follow loss then win then divide them

Incorrect Explanations for C

Answer C was the most popular incorrect answer. Again, I feel the misconception evident here is down to interpreting both the question and the tree diagram itself. What does it mean to win at least one game?

If he loses first and wins second 0.4 x 0.6 = 0.24 + If he wins first and wins second 0.6 x 0.4 = 0.24 ------ 0.48

I think this is the answer because it would either be 0.6*0.4 or 0.4*0.6 so if you add the answers together you would get 0.48.

If you go across the first win and second lose you get 0.6 x 0.4 = 0.24 And then do that for the second branch to again get 0.24 which equals 0.48

Multiply together 0.6 and 0.4 and 0.4 and 0.6. I then add the answers together after this which gets me 0.48

Incorrect Explanations for D

And then we have answer D. Now, I did not write this question, and to be honest I could not see why students would choose D. But a fair few did, and their reasons were rather interesting:

because 0.6/ 2 games = 0.30.