Lesson video

Hi, I'm Mrs. Wheelhouse, and welcome to today's lesson on comparing representations of outcomes for more than two events.

This lesson falls within our unit on Probability: Possible Outcomes.

Now, I adore probability, so I'm really looking forward to doing today's lesson with you.

Let's get started.

By the end of today's lesson, you'll be able to identify the same outcome across all the different representations.

Now, here are some key words that we're going to be using in today's lesson.

These words should be familiar to you, but if they're not, feel free to pause and read through them all.

If these words are still not familiar to you, then I suggest you go and look at some of the lessons that precede this lesson in this current unit because that's when they were first introduced.

And you may find that that helps you to understand what these words are.

But if you're ready for this lesson, then keep watching.

Today's lesson has two parts to it, and we're gonna begin by using different diagrams for multiple events.

Sophia will spin this spinner and consider different outcomes that the spinner could land on.

Now, Sophia really likes outcome tables.

She finds them good for categorising the outcomes of this spinner into two events.

So here's her outcome table.

So this one is considering whether or not the outcome is a multiple of three and whether or not the outcome is a multiple of five.

Now, Jun says, "But what if you also wanted to categorise the outcomes by whether they're odd or even? What would the outcome table look like?" Now, in previous lessons, we'd said, we think this is going to be really difficult to draw, and perhaps it might not even be possible.

Let's investigate Now if that's the case.

See, Sophia's not sure either.

"The rows are showing one event and the columns are showing another." What do we do about a third? Now, outcome tables can only categorise outcomes into two different events, and at the moment we can't see a way to put a third event in there.

Outcome trees, however, can do this with three layers of branches, it's really easy for them to show three events.

An Venn diagram just needs three circles, one for each event.

So that's quite easy as well.

To transfer an outcome to a different diagram, you have to look at where it's represented on the current diagram.

So let's consider some examples.

The outcome of 15 on the outcome tree, we can see that that's odd.

It's a multiple of three and it's a multiple of five.

So where do you think it's going to go on the Venn diagram? Let's see, well, it's odd, so we know it's going to be within this circle.

We know it's a multiple of three.

So now we're going to shrink the region that it could be in.

We know it's got to be in the circle frame three and in the circle for odd, and it has to be a multiple of five.

So that means the 15 has to go in the intersection of all three events, and that's there right in the middle.

Let's consider the outcome a five on the Venn diagram.

It's odd, it's not a multiple of three.

Can you see, it's not within that circle, but it is a multiple of five.

So what we can do now is follow the branches of our tree diagram to work out where five will go.

Remember, it's odd.

Then we have to choose the branch for not a multiple of three.

And then the branch that it's a multiple of five, and that's where the five goes.

Your turn now, where would the outcome of 30 from the Venn diagram go on the outcome tree? Would it go in space A, B, C, or D? Pause the video now while you make your choice.

Welcome back, which one did you go for? Well, we can see that 30 is both an outcome of a multiple of three, and the outcome, if it's a multiple of five, but we can see it is not odd, 30 is even.

So we need to go down the even branch, the multiple of three branch and the multiple of five branch, which means that 30 has to be where B is.

Well done if you got that right.

What about this question? Can this outcome tree showing the events of an outcome being odd, square, and greater than 10 be shown on this Venn diagram? So which of the options do you think it is? A, B, C, or D? Remember, you can pick more than one if you want to.

Pause the video and do this now.

Welcome back, which ones did you go for? Well, you could have said A, because this Venn diagram has two circles only, and we've got three events, so we do need another one.

You might also have picked D because we could show it on this Venn diagram, but only if we add another circle and it has to intersect as well.

So if you picked just A or just D, you are correct, but if you picked both, I understand why, you're making sure you're covered.

Because you could be saying, well, I dunno if I can change the diagram, or you could be saying, I might be able to change the diagram.

And that's fair enough.

Which of these Venn diagrams is able to show the events from this outcome tree? Remember, you can pick more than one if you want to.

Pause the video and do this now.

Welcome back, were you able to decide? If not, here's a quick hint for you.

Is there an outcome that works for all three events? If there is, it really narrows down which ones you had to choose.

So if you're looking your answer now and this hint supports what you've put, then you're probably right.

So did the hint help? You should have realised that you need to pick B and C because there is an outcome that works for all three events.

We can see that it's possible to be odd, square and greater than 10.

But even if we want to consider the alternative events, even, not square, less than equal to 10, there's still an outcome for that.

In fact, there's an outcome for every single possible combination of events, which means we need to have three intersecting circles.

That means it's C, but don't forget, although Venn diagrams are normally drawn circles, they don't have to be.

So B is perfectly valid too.

Let's consider this one.

Which outcomes from this outcome tree go in the locations of A, B and C on the Venn diagram? Pause while you work this out.

Welcome back, so let's consider what's going to go in each of these locations.

So for A, it was 25, remember you needed to be odd, square, and greater than 10, and the only outcome that's listed there at the end of the tree diagram is 25.

For B, it had to be odd only.

So you needed odd, not square, less than equal to 10, so that's five.

And then for C, you had to B, not odd, not square, and less than equal to 10.

In other words, even not square, less than equal to 10.

And that's two.

It's now time for your first task.

And for question one, I'd like you please to use the correctly completed outcome tree to fill in the incomplete Venn diagram.

Pause and do this now.

Welcome back for question two, use the correctly completed Venn diagram to label and fill in the incomplete outcome tree.

Pause and do this now.

Welcome back, for question three.

The Venn diagram and the outcome tree both represent the same set of events for a spinner fill in all the missing outcomes for both representations.

So you'll need to use one to complete the other and vice versa.

Pause and do that now.

Welcome back, question four.

Whilst one outcome table cannot show three events from a trial, two outcome tables can.

So what we've got here is an outcome table where we have heads and we have win or lose, and then we have red and not red.

And then we have an outcome table where we consider getting tails and we might win or lose and we might get red or not red.

So what I'd like to do is to fill in all missing outcomes on both pairs of outcome tables and then the Venn diagram.

Pause and do this now.

Welcome back.

Let's go through some answers.

So for question one, you had to use the correctly completed outcome tree to fill in the incomplete Venn diagram.

Now you'll need to check that you have put your outcomes into the correct regions of the Venn diagram.

So do feel free to pause the video at this point so you can check what I have on the screen to what you have in front of you.

Pause and do this now.

Welcome back.

Let's look at question two.

So here we had to use the correctly completed Venn diagram to label and fill in the incomplete outcome tree.

Now this is just an example because I chose to put has a digit of two as the first event.

So in other words, it went with the first layer of branches.

And the factorial event on the second layer of branches.

But you may have put these round the other way.

Equally, I chose to put has a digit as the top event and not as the bottom event.

And you might have switch these round two.

So do follow your tree branches along to see which outcome you've got and then check that by following the same branches on my diagram to see if the two outcomes match.

Feel free to pause the video right now while you do this.

Let's look at question three.

So the Venn diagram and the outcome tree both represent the same set of events for a spinner and you had to fill in all missing outcomes for both representations.

So we'll start with the Venn diagram.

In the region that considers an outcome being a factor of 360, but nothing else you had to put a three.

So in other words, the three needed to go with the 180.

24 had to go in the intersection of all three events, so it's in the middle and 16 had to go where the event has two digits and the event multiple of eight crossed over, but it cannot go in the factor of 360 because 16 is not a factor of 360.

For the outcome tree you had to fill in the missing values there and they were 90, 61, 180 and 104 Do feel free to pause there if you need a bit more time to check this.

And lastly, let's look at question four.

So you had to fill in all missing outcomes on both the pair of outcome tables and the Venn diagram.

For the outcome tables we should have for heads where we have red and lose, we have Jacob, and for not red and win, we have Jun.

On the tails table for lose and red we have Andeep.

And for win and not red, we have Izzy.

On the Venn diagram we just had to put in Laura, Lucas, Aisha, and Sam's names and you can see where they should go on the Venn diagram.

Do feel free remember to pause if you need more time to check.

It's time to look now at the second part of the day's lesson, which is on multi-stage trials with lists and outcome trees.

"So," says Jun, "when we're categorising three events from a trial, we can use Venn diagrams, outcome trees." "Ah" says Sophia, "but can we still use both diagrams when we're trying to systematically list outcomes from a three stage trial? Such as flipping this fair coin three times?" It is possible to use an outcome tree to systematically list all outcomes from a three stage trial.

Each layer of branches represent a stage, so it's possible to add even more layers if you wanted to.

So if you had a four or five stage trial.

Remember stage one would be the first flip of the coin and you can get heads or tails.

Stage two is the second flip.

So for each outcome from the first stage, we need an outcome that's possible for the second stage.

So we have heads and tails and then heads and tails again.

And this process is repeated for stage three.

The sample space created from the outcome tree is a list of all possible combinations from the three coin flips.

And you can see that here.

If you're not sure, feel free to pause and just follow the branches along with your finger.

And you'll see that the letters that you went past, IE, the outcomes that you went past are then summarised in the sample space at the end of any of the paths you took through the outcome tree.

Now it's not possible to use a Venn diagram to systematically find all the outcomes of a three stage trial.

This is because a Venn diagram is more helpful when categorising outcomes that are already known.

"Oh," says Jun, "So you can only show three stage trials using an outcome tree?" And Sophia points out, "It's always an option to just write a list of outcomes as long as you can do it systematically." And we did see that you could use an outcome table, but we had to use two to make it work.

It doesn't seem very efficient.

"Right," says Jun, "if we consider outcome trees or just listing, which do you think is better?" What do you think? Pause the video and discuss that either with the person you're sitting next to or someone at home.

Well, Sophia's view is that she doesn't think there's a right answer.

It'll depend on the question.

We should practise with both because sometimes it will be better to have an outcome tree and sometimes it will be more efficient to list them.

So for this check question, you should write a list of outcomes and then complete the outcome tree.

So in other words, do both approaches.

Now if you have a friend nearby, ask them to do one of these whilst do you do the other? That's the benefit.

You can halve the work.

So each spinner will be spun once and the coin will be flipped once.

Write down all outcomes for this trial by writing a list and then filling in the outcome tree.

Unless of course you've managed to get a friend to help, at which point you can do either the list or the outcome tree and they can do the other.

Pause now while you do this.

Welcome back, so here's our outcome tree, and at the end I've got the sample space.

Now the sample space is also the list of outcomes.

So you can check if you've got your list correct as well.

Which did you find better here? Now Juns gonna consider writing a list of outcomes for a three-course meal.

And he started, but now he's gotten stuck.

"Is it possible to use an outcome tree to complete this list and fix any errors?" Now, if we want to create a sample space of all outcomes from an incomplete list such as we have here, you have to assume that every outcome of each stage is already mentioned somewhere in the list.

If it isn't, you're not gonna know about it.

So there's no way you could possibly include it.

So for example, we have to assume the only two starters are S and G.

So I'd just come up with some examples here that S might stand for salad and the G might stand for the dumplings.

The first layer of branches will represent the two starters, and as there are only two, there'll only be two branches.

And I'm gonna put G and S by them.

The second layer of branches will be the three mains.

I can tell there are three because I can see three different outcomes listed in my incomplete list.

So remember I have to assume those are the only three because they're the only ones mentioned.

So I've got my three branches for each of G and S and I label them C, P, L.

And then the third layer represents the two desserts.

So again, each branch needs to have two branches coming out of it, and we have I and B listed on all of them.

It is possible to take an incomplete list of outcomes and create a sample space of all outcomes using an outcome tree.

And we've seen this before.

Remember, we move along the branches to generate the outcomes.

So what we could do now is compare our sample space to our incomplete list and find which ones we're missing.

Now this list shows a partially completed list of outcomes for a three stage trial.

Sophia wants to show this trial on an outcome tree.

How many branches should the first layer of the outcome tree have? Pause the video while you write this down.

Did you spot it should be four? We can see four different outcomes here.

So therefore we have to assume four branches.

What about our second layer? Now carefully read the question here.

How many branches in total will there be in this second layer? Pause the video while you work this out.

Did you spot that for each of the branches, A, B, C, and D I'll need two branches coming from them.

So that means there are eight in total.

It's now time for our final task.

For question one, a three course meal is made from one starter, one main, and one dessert.

By creating an outcome tree list all possible three course meals.

For question two, A two course meal is made from either one starter and one main or one main and one dessert.

Create two lists of outcomes to find all possible two course meals.

And then question three, how many more three course meals are there than two course meals? Pause the video now while you work on this.

Welcome back.

Question four.

Izzy has started to write down a list of outcomes for a three stage trial.

Each outcome from every stage of the trial is represented at least once in Izzy's incomplete list.

Please draw and complete an outcome tree to create a sample space of all outcomes.

Pause and do this now.

Welcome back.

Let's go through some answers.

So for question one here is our outcome tree and the sample space that generated and you should have got 18 outcomes.

Now I've listed them here and thanks to the outcome tree, they're listed systematically, so it should be quite easy to check.

But do feel free to pause if you need time to check that you've got all the same outcomes.

For question two, a two course meal is made from either one starter and one main or one main and one dessert.

You can see here that I've created the two lists and because there were only two courses here, it was a lot easier to list.

And you can see here that there are a total of 15 different outcomes.

So question three, you had to tell me how many more three course meals there were than two course meals.

Well, if there were 18 three course meals and 15 two course meals, then there were three more three course meals.

For question four asked you to draw and complete an outcome tree to create a sample space for all outcomes based on the incomplete list that Izzy had.

So you should have got 24 outcomes.

There were three outcomes for the first stage of the event, and they were one, four and seven.

There were four outcomes for the second stage of the event, and that was A F, K, and R.

And then there were two outcomes for the third stage of the event, and that was H and T.

Do you feel free to pause if you need more time to check this? Let's sum up what we've done today.

Outcome trees with three layers of branches and Venn diagrams with three circles can both be used to categorise the outcomes of a trial into three events.

Now, whilst an outcome table cannot be used to do this, a pair of outcome tables definitely can, and we saw that today.

However, Venn diagrams cannot be used to systematically write the outcomes to a two or three stage trial.

Now lists and outcome trees can be used to systematically write the outcomes of multi-stage trials and an outcome tree can be used to complete an incomplete list of outcomes.

Well done.

You've done a great job today.

And I look forward to seeing you throughout any of our future lessons on probability.