# Probability Trees TN

### Teachers Notes

This activity is about bridging the gap between the intuition of sample space diagrams and the efficiency of tree diagrams. Students will look at a problem from the two points of view, play with multiplying and adding fractions and hopefully see how tree diagrams are a more efficient way of doing the same thing! The activity is good for group work and physical manipulation, although it could be completed on computers by individuals if required. It may well take 2 to 3 hours to complete all of the tasks, but at the end, the hope is that students have a strong understanding of how tree diagrams work that they can apply to different problems.

### How

The following is some practical advice about how the activity might be run.

#### Resources provided

The activity page links to the 5 Google docs that are required for this activity. For part 2, I think it is best if the fractions are printed and cut out for work in groups of two or three. It is just nicer to be able to move things around a bit. For Part 4, the tree diagram might be better blown up to A3 for each of the groups. Here are the solutions to the multiplying exercise and also shown in the context of the tree diagram.

For the addition exercise, the 9 fractions represent the 9 probabilities associated with the 9 different outcomes from the tree diagram. The first three are the probabilities of the three outcomes that are the same colour,

\(P(RR)=\frac { 9 }{ 100 } ,\quad P(BB)=\frac { 1 }{ 25 } =\frac { 4 }{ 100 } ,\quad P(GG)=\frac { 1 }{ 4 } =\frac { 25 }{ 100 } \quad \)

\(They\quad add\quad up\quad to\frac { 19 }{ 50 } ,\quad the\quad others\quad add\quad up\quad to\quad \frac { 31 }{ 50 } \)

#### Resources needed

As suggested above, the activity goes well in groups of 2 or 3 with cut out fraction cards and a blown up version of the tree diagram in part 4.

Time needs

This activity could take up to 3 hours in total, as always, depending on the nature of the group. I have used it as a way to get to grips with the mechanics of a tree diagram, whilst relating it to sample space diagrams and feel it is worth the time investment. The activity does require some review of adding and multiplying fractions and, in my experience, this can take some time!

#### Starting and finishing

- Pose the problem set out in 'Part 1' and get students to commit to a 'bet'. It can be fun to have something to bet with here.
- Complete the tasks in part 1 and pay out on the bets! This is a good way of getting students to reflect on their instincts against a theoretical analysis. It is a fun discussion to have, if time allows, on what the limitations of our instincts are where probability is concerned.
- Part 2 - This is a review of fraction manipulation. Give out the cards and start the task of making 9 groups of 3. Students are then invited to consider how these fractions relate to the original problem.
- Part 3 - As above, only with adding now.
- Part 4 - This is the big part, where students can see the link between the sample space and the tree diagram. The tree diagram can be filled in using the sample space diagram, but when it is finished, the multiplications and additions from the previous activities can be seen on the tree diagram. There is an opportunity for some significant discussion and demonstration here to help students see the link.
- Part 5 - This is of course, the acid test of whether or not the concepts have been understood! Students are helped by being given some of the fractions involved.
- After this, it is probably a good idea to serve up some questions from other sources to help students apply what they have learned in different contexts.

#### Records

Up until 'Part 5' students may well have been working in groups and so will not have their own versions to take away. For this part I have found it helpful to take photographs of what they are doing for records. These or, a hard copy from one group, can be displayed in the classroom. If you keep a class blog or something like that then these pictures can be posted there for easy reminders.

### What to expect

The longer an activity is and the greater the degree of 'discovery' involved, the more potential there is for things to go off track! This is not a good reason not to try it though!

- Firstly, the most important thing to deal with is the 'semi-reality' of the counters in bag scenario. This is easily overlooked in a culture where using
*real*, real contexts is talked about so much. We all run our classrooms differently, but I think it is important to spend some time exploring real applications of probability and then explaining how we can learn to understand those concepts through 'semi-real' models. Of course, if students do bet something real and do stand to win something if they win, then the context becomes real. An idea would be to give them all sweets to bet with and promise to double it if they win or take it if they lose. - Parts 2 and 3 involve remembering how to add and multiply fractions and understanding equivalent fractions. Although, these are often considered 'basic' skills, I find the opposite to be true and that these concepts need regular revisiting. It need not take too long, just expect to have to deal with it.
- Part 4 - as suggested above, students can fill the tree diagram in from the results of their sample space diagram. The real 'wow' moment can come when students recognise the multiplications and additions they have done along the branches of the tree diagrams. This is easily missed and thus the main point of the activity is missed! Take some time to make sure students are looking at this.