One topic that I used to get very frustrated with teaching is ‘probability trees’. It would frustrate me for the same reason that topics such as ‘angles in parallel lines’ (which I’ve previous written about here) would frustrate me: it’s relatively easy compared to other parts of the maths curriculum, but students seemed to struggle with it more than with difficult topics. Even when I’ve taught top set Year 11, who could confidently solve non-linear simultaneous equations and complex trigonometry problems, I’d be surprised that probability trees would be one of their most requested topics for revision.
Looking back over my old teaching resources on probability trees, one possibility for these difficulties could be that I didn’t invest enough lesson time on the aspects of the concept that were most novel to them. Take the question below for example.
Questions such as this one require students to:
- know how to write probabilities of single events as fractions;
- know how to multiply together fractions;
- know how to add together fractions;
- know how to structure a probability tree to represent a sequence of possible events;
- understand why we multiply the fractions together for combined events.
The first three bullet points are things that students (hopefully) already know before getting to this section of their programme of study. This means that when students are introduced to probability trees, the most unfamiliar aspects of the topic are the last two bullet points.
Thinking back to those old lessons, a lot of my explanations would centre around solving whole probability tree problems from the get-go. While solving a problem on the board, I would call on students to give me the fractions that go on each line, multiply together each pair of fractions and add up the fractions that I needed to solve the problem. The students would answer these questions so confidently that I’d think, “They’ve got this!” and set them off with independent practice. I’d then always be surprised with how difficult they found solving probability tree questions by themselves, how many marks would be dropped in exams on them and how it would often be a common request for revision.
The biggest issue tended to be drawing the tree! Knowing how to set them out, how many branches to use, how many sets of branches, how to arrange the labels, etc. So even though students knew how to perform all the calculations they needed to solve a probability tree question, they made mistakes before getting to the point of calculatuon. I think the biggest issue was that some of my students didn’t seem to understand what was going on when they were drawing probability trees and what the diagrams really meant.
Evidence of such misconceptions would occasionally present themselves in the way students set out their branches…
…the way they arranged the information on trees…
or their choices of fractions in problems that were worded like the one below.
In hindsight, I suspect that some of the students’ uncertainties and mistakes with this topic were because I tended to skim over explanations about how to set out probability trees and their meaning. Instead, I jumped too quickly to calculations and solving complete problems. This meant that more time was spent discussing the aspects of the problems that students were already familiar with (how to probabilities and calculate with fraction) than the aspects that were most novel to them.
This blog post shares a selection of things that have gradually been added to my lessons on this topic over the years. Some of them support the introduction of probability trees by helping students think about their structure and meaning; some help extend and challenge students further within the topic. I don’t always use them all every year; it usually depends on how much support of extension I feel is appropriate. I’ve also discussed some of these things in conversion with Craig Barton on his podcast (available here).
Laying Some Early Foundations
Students can start to be familiarised with some ideas relating to probability trees long before they get to the ‘probability tree’ section of the scheme of work. For example, students may work with frequency trees in younger years to solve problems like the one below.
The numbers above were modelled roughly on the probability tree from the question below. While the frequency tree looks at observed data for many people playing the games, the probability tree looks at the theoretical probabilities for each individual person playing the games.
Inspiration for this connection was taken was taken from a series of interesting articles on the Nrich website about transitioning from frequency trees to probability trees (available here). They describe an empirical-to-theoretical approach for probability, where students go from exploring frequency to proportions to expectation.
Another thing that I’ve found useful, as a way provide students with some early exposure to probability trees, is to represent outcomes of single events with branches during lessons on basic probability. These don’t necessarily change the way I teach probability to Key Stage 3 or affect how students use probabilities. They are simply used to as a way to supplement explanations with illustrations on the board. I find that if students are at least familiar with seeing outcomes and probabilities represented this way for single events, then they are already one step ahead when it comes to combining them to build probability trees.
Thinking About How to Structure a Probability Tree
My go-to tool for introducing various ideas within this is to use scenarios where someone is drawing different coloured marbles from a bag. While these ‘marbles-in-a-bag’ scenarios are fairly contrived, they provide a helpful stylised way to explore aspects of probability trees because they are easy to visualise and play out if necessary. So, at each stage of concept development, I often begin with a marbles-in-a-bag scenario first and then later ask students to apply what they have learned to a more contextual scenario.
The first thing I tend to focus on is how to choose an appropriate tree structure to match a sequence of events. This is without writing any probabilities for the time being. For example, we might start by considering how to draw trees for the four scenarios below.
By comparing and contrasting these scenarios, students can be encouraged to consider the similarities and differences between the four probability trees.
It can be helpful for students to explicitly consider and discuss some of the key questions in their decision making. “How do we know how many branches to draw at each intersection? How do we know how many layers of branches to draw?” Once students have understood the reasoning behind the structure for these four probability trees, they might then apply what they have learned to draw trees for other contexts.
I tend to find that Scenario C causes the biggest debate with students. They can sometimes be tempted to draw two layers of branches for ‘on time’ and ‘late’, with three branches at each intersection (one for each person). This is similar to the mistake at the start of the post. This task provides a chance to address the issue before students meet it when solving complete problems.
One useful follow-up to this activity can sometimes be to take the contextual scenarios and map them to the marble scenarios. “Why do the probability trees for ‘Marble Scenario 4’ and ‘Contextual Scenario A’ have the same structure? Which of the marble scenarios has the same structure as Contextual Scenario C? How could I alter one of the contextual scenarios to make it have a tree like Marble Scenario 3?”
In all of the examples above, the conditions are consistent throughout the tree. For example, if one intersection in a tree has three branches, then all intersections have three branches. So, it can also be valuable to explore ways that the number of branches can vary within a tree.
In the scenario below, we are choosing marbles from two different bags. Students can consider the different ways that we could structure our probability tree.
They might reason that the number of branches per intersection differs between layers is because the number of different options differs between picks. This could also occur when drawing two marbles out of the same bag, when there is only one marble of a certain colour. For example:
In the scenario above, students can observe how the outcome of one event affects the number of possible outcomes in the next event. This highlights that some branches can terminate before others.
Other contextual situations where branches terminate like this could be playing in a knockout tournament (e.g. there are four rounds of matches and you play until you lose) or in Monopoly where you have three attempts to roll a double to get out of jail without paying.
While all the examples above start by describing a scenario and then asking students to think about the tree, it can also be interesting to do this the other way around. Students could be presented with a completed probability tree and then be asked to think about the scenario.
Once students are comfortable setting up trees of different structures, it doesn’t take much more effort for students to start labelling branches with probabilities.
Thinking About Why We Multiply Probabilities
Another novel aspect of this topic is the idea of multiplying two or more probabilities together. Students have most likely solved problems before where they needed to add probabilities together (i.e., when finding the probability of one thing happening or another) and hopefully understand why they should use addition. But this could be the first time that they find themselves multiplying probabilities together and trying to understand why they should do so.
One way that I like to get this across is by referring to their previous experience with using two-way-tables as sample spaces. For example, students could represent the scenario below by using both a probability tree and a table.
By comparing the two diagrams, students can consider how to find probabilities for combinations of events. Students tend to see clearly from the table that the fraction of BB is four twenty-fifths. They might then spot that they can also get four twenty-fifths from the tree by multiplying the two branches together. Digging a little deeper, they might reason that this works because the table shows that two-fifths of the outcomes have blue as the first pick (10 out of the 25 boxes) and then two-fifths of those also have a blue as the second pick (4 out of all 25 of the boxes). Therefore, they are finding a fraction of a fraction, which can be done by multiplying the two fractions together. Once this rule has been established and checked then we can put aside the two-way table and focus on solving future problems using just probability trees.
Reasoning and Problem Solving with Probabilities
Once students get to the point where they understand how to structure probability trees and can use them to calculate probabilities of combined events for both independent and dependent cases, it can be tempting for to think “job done!” and move on to the next topic. This is certainly where my lesson resources used to end (and occasionally still do). In most cases, this often enough to enable students to solve most probability tree problems at GCSE. However, there is so much scope to extend and apply probability trees within the GCSE syllabus.
One avenue of reasoning could be to explore how the order of events can affect probabilities depending on whether they are dependent or independent on each other. Take the problem below, which asks students to consider two independent events.
Some students might be hesitant to set up the tree for a problem like this because the wording is a little vague. They might be more used to scenarios where an event is repeated twice, or two different events happen in an obvious order (e.g., Monday and Tuesday). However, the wording in the scenario above doesn’t give any clues about what order the boxes are chosen from. This vagueness though provides opportunity to pose questions such as: “Does it matter which box Leyland chooses from first? Will the order affect the probabilities of the combined outcomes?” It can be good to get students to explore this through verbal reasoning first, discuss how the fractions will be the same in either order and how multiplication is commutative. These ideas can be confirmed by drawing probability trees for each order.
But then what if we made some adjustments to the problem to make it so that the probabilities for one event depended on the outcome of the other? For example, there have been problems in GCSE over the last few years where a marble is taken from one bag and paced into another.
In the image below, the same problem is presented twice but with the order switched. Similar questions could be posed again: “Does it matter which box Laura chooses from first? Will the order affect the probabilities of the combined outcomes?”
Once again, it can be interesting for students to explore this through verbal reasoning before drawing any trees. They may recognise that the fractions will be different but could still have some doubts about if it matters. “What if the fractions are different but they multiply and simplify to give the same products?” Some students I’ve taught have managed to convince others that the order does matter by focusing on the denominators: in the left-hand scenario, the denominators will be 5×8=40; in the right-hand scenario, the denominators will be 7×6=42. Students predictions can be confirmed by drawing the probability tree for each case.
There are also plenty of opportunities to introduce other aspects of the mathematics curriculum into probability trees. One way could be to include other topics within the context of a problem, such as in the question below. While this scenario is fairly contrived, it requires students to draw upon their knowledge of area, perimeter and factor pairs in order to calculate the probabilities that go on each branch.
The context of the problem isn’t the only way to weave in other topics. There is scope for students to exercise other maths skills even within more abstract probability tree questions. Take the one below for example. This could be solved numerically through trial and error or algebraically through equations.
However, what I particularly like about this problem is its scope for extension. A follow-up problem could be to say, “When I wrote this problem, I chose to use the terms x, 2x, 2x, and 4x and it worked out quite nicely. Would it have worked just as well if I had chosen a different set of four terms? For example, what if I change this 4x to 3x? Would the problem still work?”
Once students start working on this, they can find that it is unsolvable. They might find values that work for some combined outcomes, but not work for all. But what if I use the terms x, 4x, 4x and 16x instead? These do work! This now seems ripe for turning into an open-ended investigation: “Find as many sets of four coefficients as you can that work in this problem.”
Students often start by choosing four terms, drawing the tree diagram and then trying to work out the fractions to determine if it’s possible – similar to the order that they solved the first of these problems in. However, they might then figure (or be prompted) that a better strategy could be to work in the reverse direction: choose a number of marbles for each colour, draw the tree and then work out the terms. Once they get into the swing of doing this, they might start using a systematic approach to choosing the numbers of marbles, which reveal patterns in the coefficients.
By keeping the number of marbles for the first colour as 1 and changing the number of marbles for the other colour, students can generate the following sets of terms:
There are interesting patterns that students might explore here on a basic numerical level here. They might notice that the middle two terms are always the same, the coefficient of the fourth term is always a square number and is the square of the second/third coefficient, all four coefficients always add up to the next square number that comes after the fourth coefficient, etc.
Students can dig deeper by changing the number of marbles for the first colour. Before doing so, it could be good for them to predict which of their observations will remain true when there is more than one of each colour. For example, when there are 2 marbles for the first colour and they vary the number of the second colour, they can get the following terms:
When there are three marbles for the first colour and the number of marbles for the second colour varies, they can get the following:
Once again, there are interesting things for students to explore on a numerical level. The coefficients still add up to a square number (but not the next square number after the fourth coefficient anymore). You can get the first and fourth coefficients by squaring the number of marbles of each colour. You can get the second and third coefficient by multiplying together the number of marbles for each colour… and so on.
If this is all that students get out of this problem, then great! However, students may also make a link between this problem and another aspect of the mathematics curriculum: squaring binomials. The patterns of coefficients in each set follow the same patterns as they do when they expand brackets for the square of a binomial.
For students who are planning to study A-level Maths, investigations like this can act as a ‘teaser trailer’ for ways that probability becomes more algebraic in the further study. For students aren’t planning to study A-level Maths, it can simply highlight a surprising connection between two very different-looking things that they have previously learned for GCSE. But even if they don’t make the algebraic connections, the numerical reasoning can help spot patterns in probability trees with repeated independent events. This might come in handy when they next solve a basic probability tree problem (such as the one below), because it provides them with some ‘checking mechanisms’ such as, “I should expect these numerators to be square numbers,” and, “I should expect these two numerators to be the same and also be product of the numbers of marbles for each colour.”
If you would like to use or share any of the images from this blog, feel free to use this Power Point to help you: Thinking About Probability Trees
Previous blog posts:
‘Planning Topics’ Series:
- This Way, That Way, Forwards and Backwards
- Filling in the Gaps With Histograms
- Twists and Turns With Straight Line Graphs
- Thinking About Corresponding Angles
- Thinking About Areas of Circles
- Thinking About Areas of Parallelograms
- Thinking About Ratio and Algebra
‘What I Learned From Shanghai’ Series:
‘Making Connections’ Series