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 the probability has to offer.

And we're gonna 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 we 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.

And we're gonna 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 gonna 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 span 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 gonna 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 span spinner one, I span 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 gonna 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 gonna spin spinner two.

What's gonna 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 gonna 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 in 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 outcome 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.

For question five, we spin spinners 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 AB and then AB.

It doesn't matter if you did that or if you swap these around, 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 question six, you had to complete the spinners.

So you need 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 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 one, two, three, four, five, and six.

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 two, four, six, and odd has the outcomes one, three, five.

Of the even numbers the spinner could land on which ones of these are factors of six? So what do you think is gonna 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 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 three is a prime number, which means one and nine 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 or 64.

Jacob's gonna spin the 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 and 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.

In 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 rounds 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.