Lesson video

Hi, I'm Mrs Wheelhouse and welcome to today's lesson on using an outcome tree to display outcomes for two events.

This lesson is from the unit on Probability: Possible Outcomes.

I love probability, so I'm really looking forward to further exploring all that probability has to offer.

And we're going to be starting in this lesson with outcome trees.

So, let's have a look at those now.

By the end of today's lesson, you'll be able to systematically find all the possible outcomes for two events by using an outcome tree diagram.

So we've got a few key phrases today.

Tree diagrams. Now tree diagrams are a representation used to model statistical or probability questions and branches represent different possible events or outcomes.

Each branch of an outcome tree shows a possible outcome from an event or from a stage of a trial.

The full outcome tree shows all possible outcomes.

A sample space is all the possible outcomes of a trial.

And a sample space diagram is a systematic way of producing a sample space.

So you might hear those words mentioned today in our lesson, and when you do, if you are a little concerned about them or you don't quite remember what they mean, feel free to pause the video and come back to this part so that you can refresh your memory.

Our lesson today has three parts to it, and we're going to begin by considering outcome trees for two-stage trials.

So spinners one and two will both be spun once.

Let's list all the possible outcomes.

So it could be AB, AB, AC, or if we got a B on spinner one, then it could be BA BB, BC.

Sofia says, "I wish there was a way of making sure I've listed all the outcomes from this trial." Now she's been systematic in how she listed them and because there's a small number of outcomes here, it's pretty clear she can be certain.

However, we are going to get onto the point where there are more outcomes and it might become more complicated.

So let's start with this situation where things are a little simpler and we can build out from there.

So Jun says, "I know you can use a two-way table, but let's break it down even more." And an outcome tree helps show the outcomes of a trial and we can treat an outcome tree as a timeline of possible outcomes.

And this might make it easier to understand.

So we begin at the point before spinner one was even spun and we say we're going to spin spinner one.

What can happen? So we spin that spinner and the two branches I've drawn here represent the two possible things that might happen.

In other words, one branch represents one outcome and the other represents the second outcome.

So for example, I could get an A or I could get a B.

So this is the full sample space for spinner one as all outcomes of the spinner are represented.

In this scenario, spinner one landed on A.

So if we followed that branch, it tells us we spun spinner one and we got an A as our result.

Now we will spin spinner two.

What outcomes are possible from spinner two? That's right, there are three, A, B, and C.

So I could get an A or I could get a B, or I could get a C.

So can you see there are three branches here? One for each possible outcome.

So let's consider the scenario where spinner one showed B.

We're now going to spin spinner two.

What outcomes are possible from spinning spinner two? That's right, it's still A, B, and C.

So we could have A, a B or a C.

We have here for the first two branches, the outcomes from spinner one.

And then on the second set of branches we have the outcomes from spinner two.

Now they've been repeated twice, and that's because one set belongs with getting A on the first spinner and one set goes with getting B on the first spinner.

And we can see that really clearly here.

Jun says, "This is a cool set of scenarios showing what could happen." I think Jun likes how nice and clear it is.

"But where does it show the outcomes from a whole trial?" So what he means is, I spun spinner one, I spun spinner two, what was the result? Well, if we traverse or move through each possible path, we'll end up seeing all possible outcomes of this trial.

So if I follow the path I can see now, I got an A on spinner one and an A on spinner two.

So my outcome is A and A.

If I follow this path, however, the outcome is A, and then B.

And then my final path for taking the top route is A and C.

What about though if I'd followed B? Well I could have gone down the path for B and then I could have got an A.

So we've got BA, there's also BB and BC, and now I can see all possible outcomes for this trial.

Here is our sample space.

Quick check now.

Spinner one will be spun once.

How many branches will there be in this layer of branches? So in other words, how many branches are going to come from our starting dot? Pause while you write down what you think.

Welcome back.

How many outcomes were there for spinner one? There were four outcomes.

So there should be four branches, one for each outcome.

So what's going to go in those two gaps? So I've started to fill in this tree diagram for you.

What do you think will go in the two gaps? Pause and write down your answer.

Welcome back.

You should have put C and D.

Now you could have put these in either order, but we do try to be systematic in how we label our branches.

So given we went AB, it would make sense to finish off CD.

But remember it doesn't actually matter, so you could have gone the other way.

Now, we've spun spinner one and now we're going to spin spinner two.

What's going to go in the gap here? Pause and write down your answer.

Welcome back.

If we've already shown the outcome of lose, then we should have to write win here because the other possible outcome for spinner two.

Now I filled in a bit more now of my tree diagram.

What does the highlighted pair of branches represent? Pause and write down which one you think it is.

So A, B, C, D, or E.

You should have chosen D, which was that spinner one landed on B and spinner two landed on lose.

Remember the first set of branches referred to spinner one and the second layer or second set of branches referred to spinner two.

A sample space showing all outcomes from this outcome tree is now made.

What's going to go in that gap? Pause and write down your answer.

Welcome back.

Well if I follow the branch through right from the start, I start at the dot, I've gone down branch C and then I've gone up the branch that says win.

So it should say C and win.

It's time now for your first task.

In question one, we're going to spin the spinner once so I'd like you to complete the outcome tree for this single stage trial.

In question two, spinners one and two will each be spun once each.

What are the two branches in the box you can see represent? And then for part B, please complete the outcome tree.

Pause while you have a go at this.

Welcome back.

Question three now.

Spinner three will be spun twice.

For part A, please complete the outcome tree and then in B, complete the sample space of outcomes for this outcome tree.

For question four, spinners four and five are spun once each and they have these outcomes.

So spinner four has the set of outcomes, A, B and spinner five has the set of outcomes red, blue.

Please complete the outcome tree and sample space.

Pause while you do this.

Welcome back.

Just two questions to go now.

Questions five and six.

Question five, we spin spinner six and seven.

We need to complete the outcome tree using the sample space please.

And then in question six, spinners eight and nine will be spun once each.

Please fill in the outcomes of each spinner from the outcome tree.

So the spinners need to be completed.

Pause and do this now.

Welcome back.

Let's go through some answers.

So question one, you had to complete the outcome tree for the single stage trial.

I chose to follow this alphabetically, so I went B, C, D, but again, you could have put those letters in a different order.

In question 2A, you had to state what the two branches at the start represented and that's the two outcomes from spinner one, then complete the tree.

So you needed to put B as the missing outcome in the box and then at the end fill in the missing outcomes.

So at the top you had to put X, but for the second set of branches down below, you just needed X, Y, Z in any order.

Question three, we spun this spinner twice.

You had to complete the outcome tree and then complete the sample space.

So you should have got the answers you can see there.

Do feel free to pause and check.

In question four, we spun spinners four and five once each and we saw those outcomes.

You had to complete the outcome tree.

Now you could have swapped the order of these, so you might have said that instead of writing A and B, you wrote red and blue there.

And then at the end of the second set of branches, written A, B and then A, B.

It doesn't matter if you did that or if you swapped these round.

What you will notice is you still get the same four outcomes.

So do check that you've got that.

Question five, you had to complete the outcome tree from the sample space.

So you had to have G at the end of the first set of branches and then it should be win, lose, draw so that your answers in the outcome tree match up to the order you can see in the sample space.

And the same for the second half down below.

And then for question six, you had to complete the spinners.

So you needed to put U and V on the top spinner.

And again, you could have put 'em in a different order and the bottom spinner needed to C, D, and E.

But again, you could have put these in either order.

It's now time for the second part of our lesson on constructing outcome trees.

You've done a bit of this, but what we haven't done is gone into it in much depth.

So let's do that now.

Jun says, "Does it matter if we spin spinner two first and then spin spinner one?" What would change if we did this? What do you think? Well, Sofia says, "Oh, I'm not sure.

Let's make our own outcome tree and find out." So we're going to construct an outcome tree where the first layer of branches represents the outcomes of spinner two.

Well let's do that together.

We're going to start therefore by drawing the dot and that represents the moment in time just before we spin the second spinner.

Remember, 'cause we're spinning spinner two first this time.

It has three possible outcomes so we need three branches.

Remember, each branch is representing a possible outcome.

So now we label them to make it clear which is which.

So we have A, B, C.

Now we have to imagine at this point that spinner two landed on the A.

And we start from this point and we spin spinner one.

We now need to draw two new branches to show the two possible outcomes that spinner one has.

So we've done that and then we do need to label them so we have an A and a B because they are the two outcomes from spinner one.

We now repeat this process by considering what if spinner two had landed on B? Well spinner one could have landed on A or B, and that's the two branches you can now see us having drawn in.

We now need to consider what would've happened if we spun spinner two and we got a C, and we need the same two branches showing the outcomes of spinner one.

So A and B.

Now to the right, we're going to draw a sample space that lists all outcomes from this trial and we're following through our branches.

Can you see how AA appears at the top in line with that top A? And it's to show me that if I'd followed from the start the top branch and then the top branch again, spinner two spun and got an A and then spinner one spun and also got an A.

So I got the outcome AA.

Next branch down would've been if I'd got A and then B.

And I can complete this for all branches.

So if you look at the screen, you can see that this sample space is from an outcome tree where spinner two was spun first.

And this sample space is from an outcome tree where spinner one was spun first.

Sofia points out that the list of outcomes look different between the two outcome trees we drew.

Well they do look a bit different 'cause I can see AC in the first sample space, the one on the left, but I can see CA from the sample space on the right.

Oh, Jun says, "I don't think so because actually we have the same outcomes here." What do you think? Drawing the outcome tree with spinner two first has just rearranged the order.

Ah, I see what you means.

So if we consider that AC again, that was me spinning A on the first spinner and C on the second spinner, whereas CA in the second sample space represents me spinning spinner two and getting C and then spinner one and getting A.

It is the same, but the order the letters appeared in have just switched round depending on which spinner I spun first.

Let's do a check.

An outcome tree is drawn from spinners one and two.

In what ways has this outcome tree been drawn incorrectly? Pause and do this now.

What did you go for? You should have said there were two ways.

So it's actually A and D.

For starters, there aren't enough branches and some of them are in the wrong place.

This is what the tree should look like.

So in what ways has this outcome tree been drawn incorrectly? And again, you can choose from A, B, C, or D.

Pause and make your choice.

Welcome back.

Which ones did you go for? You should have picked C and D.

Once again, the branches are in the wrong place and the second layer of branches are not lining up with the first layer at all.

This is what our tree should have looked like.

It's time for our second task.

In question one spinners one and two will be spun once each.

Please complete the outcome tree and sample space.

And in question two, spinner three will be spun twice.

Please complete the outcome tree and sample space.

Pause while you do this now.

For question three, spinners four and five has spun once each and have the following outcomes.

So spinner four has the outcomes, A, B, C, D, and spinner five has the outcomes on and off.

Please draw an outcome tree and list all outcomes in a sample space.

In question four, Izzy spins the spinner with the outcomes win and lose.

If the spinner lands on win, she stops but if it lands on lose, she spins once more.

Draw an outcome tree and list all outcomes in a sample space.

Pause and do this now.

It's time to go through our answers.

So for question one, spinners one and two will be spun once each and you had to complete the outcome tree.

And you can see here how I've completed mine.

For spinner three we spun this one twice and I said to complete the outcome tree.

And again, you can see how I've done this on the right-hand side of the screen.

Do feel free to pause while you check your work.

Then for question one and two, you also had to complete the sample space.

Feel free to pause and check that your sample space matches mine.

Remember you may have these in a slightly different order if for example, you'd swapped B and C around or J and K around in the first question.

For question three, asks you to draw an outcome tree and then complete a sample space.

So you should have all these possible outcomes.

And then Izzy spins a spinner with the outcomes win, lose.

If it lands on win, she stops but if it landed on lose, she spun once more.

Remember only once more so she doesn't have to keep spinning every time she gets lose.

So you should have an outcome tree that looks like the one you can see at the bottom of the screen on the right-hand side.

And then our sample space would be win and then that's it 'cause she would've stopped or lose, win, lose, lose.

It's now time for the final part of our lesson, which is considering outcome trees from two events.

Outcome trees are also very useful in categorising all of the outcomes of a trial into different events.

So this spinner will be spun once.

The possible outcomes are 1, 2, 3, 4, 5, and 6.

Now the spinner could have landed on either an odd number or an even number.

So we could think about this as we get an even number or an odd number, where even has the possible outcomes 2, 4, 6 and odd has the outcomes 1, 3, 5.

Of the even numbers the spinner could land on, which ones of these are factors of six? So what do you think is going to go by the factors of six? That's right, the two and the six.

And therefore the four goes with not factors of six.

We can again use that event and consider what would happen if I'd spun odd first, which of those odd numbers are factors of six and which are not? So what am I going to write by factors of six? That's right.

One and three.

And by not factors of six, that's where the five goes.

So event one member was, is the result even or not even? And the second event was, is the value a factor of six or not? Two and six are the outcomes of the two events that the spinner lands on an even factor of six.

Whereas five is the outcome of the two events that the spinner lands on an odd number that is also not a factor of six.

When using an outcome tree to group a single trial into different events, each event can be split into one event and its opposite event.

So that's how those two things didn't overlap.

Quick check.

This spinner will be spun once.

An outcome tree shows the event the spinner will land on a factor of 18.

What are the outcomes for this event? Pause while you write 'em down.

Welcome back.

You should have filled in one, three, and nine because they're factors of 18.

What is the opposite event to landing on a factor of 18? Pause and write it down.

Welcome back.

You should have filled it in with not a factor of 18 because that is the opposite of getting a factor of 18.

Now I've also filled in the two outcomes that go with that.

So that's 7 and 25.

So what I'd like you to do now is place the outcomes on the correct events.

So now we're considering whether or not our outcome is a prime number or not a prime number.

So by A, B, C, and D, fill in what the outcomes will be for each of those places.

Pause and do this now.

Welcome back.

Let's see what you filled in.

So of the factors of 18, 1, 3, 9, only 3 is a prime number, which means 1 and 9 both had to go by B.

For the two numbers that were not factors of 18, so 7 and 25, 7 is a prime number, so goes by C and 25 is not, so goes by D.

Now the spinner lands on a number that is not a factor of 18, but it is a prime number.

Which number did the spinner land on? Pause and write that now.

Welcome back.

Did you say seven? You should have done because remember we went down the branch that refers to not a factor of 18 and then the branch that refers to a prime number, and seven's the only outcome listed there.

It's now time for our final task.

We're going to spin this spinner once, complete the outcome tree by showing where the different outcomes go.

In part B, the spinner lands on 11.

Using the outcome tree, which two events just took place? Pause while you complete this.

Question two, a spinner has outcomes of 5, 9, 15, 25, 28, and 441.

Now I'm asking you to complete this outcome tree and you might think you don't have enough information, but if you look carefully enough, you'll see in fact that you do.

So slightly tricky this time.

But pause the video, I know you can do this.

Good luck.

Welcome back.

Question three, a spinner has outcomes of 1, 4, 8, 16, 25, 27, 30, 36, 39 and 64.

Jacob's going to spin this spinner and he looks for the events of landing on a square number and also a cube number.

So please draw and complete an outcome tree for this trial.

And then in part B, how many outcomes match the events that Jacob is looking for? Remember, Jacob wants a square number that is also a cube number.

Then in question four, we have a cafe where the following drinks are on the menu.

I'd like you please to create an outcome tree with the first event being has a handle and the second event being has a straw.

And then on your tree, circle the outcomes that represent straw, but no handle.

Pause and do this now.

Welcome back.

Let's go through some answers.

So, for the outcome tree, I filled in where the different outcomes go.

You'll notice that there were no square numbers that were also prime numbers.

Did that surprise you? I suspect not because you know that a prime number can only have two factors, one in itself.

In part B, the spinner lands on 11.

The outcome tree shows you which two events must have happened.

And we must have had the event, not a square number, but also the event prime number.

Question two, you had to complete the outcome tree.

So please feel free to pause the video if you want to check your answers against mine.

You did have enough information to fill this in because you knew one event was multiple of five, the opposite or the complement event had to be not a multiple of five.

And for the second phase we could see not a square number so we knew the opposite was square number.

And question three, we had our list of outcomes and Jacob was going to spin the spinner and look for events of landing on a square number that is also a cube number.

And there were two outcomes that satisfied these conditions, and that's 1 and 64.

For four, you had to create an outcome tree with the first event being handle and the second has a straw.

And I've got an example of a tree diagram here.

Remember, you might have swapped these round so no handle might have been on the top branch and handle might have been on the second.

But you'll still get the same outcomes.

For B, you had to circle the outcomes that represent having a straw with your drink, but no handle.

So you should have circled the outcomes D, E, and H, and they should have all been together.

It's now time to sum up today's learning.

The branches of an outcome tree show outcomes of a trial.

And when looking at two events from the same trial, a tree diagram helps to organise the sample space according to those two events and their combinations.

When looking at a two-stage trial, the tree can help find the sample space in a systematic way by considering the two parts of the trial and their outcomes one at a time.

A sample space can be made from any outcome tree that represents multiple stages of a trial.

Well done.

You've done a fantastic job today.

I hope you found learning about outcome trees useful and that you could see how they're an alternative way to completing a sample space.

How did you feel about them? Hopefully you enjoyed it.

I look forward to seeing you for more lessons on probability.