Lesson video

Hi, I'm Mrs. Wheelhouse and welcome to today's lesson on using a Venn diagram to display outcomes for two events.

This lesson falls within our unit on probability possible outcomes.

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

Let's get started.

By then of today's lesson, you'll be able to systematically find all the possible outcomes for two events by using a Venn diagram.

Let's get going.

So the key phrase we're using in our lesson today is Venn diagrams. Now, Venn diagrams are a representation used to model statistical or probability questions.

Now commonly we use circles to represent events, but it doesn't have to be a circle.

And as you'll see in this lesson, we don't always use a circle.

Sometimes the shape we use could be a bit more like an oval, a rough circle, so you could look at it and think that wasn't drawn with a compass.

So it's important to know it's roughly a circle is what we tend to use, but we don't have to be super precise.

Now our lesson comprises of two parts, and we're gonna start with the first part on Venn diagrams. When this spinner is spun, the possible outcomes are 1, 2, 3, 4, and 5.

And we refer to this as our sample space.

An outcome table can be set up to categorise the possible outcomes.

So in this case we're gonna consider whether an outcome is odd, and we're gonna consider if an outcome is prime.

And you can see where we filled in the numbers one to five in our outcome table.

We might also use an outcome tree.

So you may have seen these before and it's great if you have, if you haven't and you want to become more familiar with them, you could go and look at the lessons on outcome trees and on outcome tables.

So you can see here we've got is a number prime or not prime at the end of the first set of branches.

And the second set of branches is looking at if an outcome was an odd number or not.

Another representation is a Venn diagram.

So let's see how we draw this.

We start by considering the sample space and we represent this with a rectangular box.

We're going to put it in our event P.

It didn't matter whether I drew P first or O first, but I'm gonna focus on event P in this particular instance and I've drawn P with a circle and you can see inside it I wrote the outcomes two, three, and five.

So what this means is the two, the three and the five are inside the circle, and that means they fulfil the conditions of that event.

So think about it, P was a prime number, so two, three and five were my outcomes that were prime and that's why they're inside that circle because they fall within that event.

You can see where these numbers came from in my outcome table.

Now let's consider event O.

So you can see here I've drawn event O on and I filled in the one, the three and the five, except the three and the five were already there.

Can you see how when I drew my second circle, I've got it overlapping my first circle, and that's because the three and the five are both prime numbers and odd numbers.

So they need to go in the space that represents both events.

And that is where the two circles cross.

Now four is not a prime number and it's not an odd number, so it can't go in either circle.

Remember, the outcomes inside the circles are ones that form the set of outcomes for those events, and four doesn't.

But it is part of our sample space.

So we do need it on our diagram and that's why it's gone inside the rectangle, but outside the two circles.

So let's do a check to see if we understand a Venn diagram.

Event A refers to even numbers and event B refers to factors of 20.

What I'd like you to do, please, is complete the sample space that shows all the possible outcomes using this Venn diagram.

Pause and do that now.

Welcome back.

What did you write? You should have the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.

Each of those outcomes appear once in our Venn diagram.

Now here we've got a Venn diagram, but we also have an outcome tree as well.

And we can see our two events.

Event A refers to an outcome being a square number and event B considers outcomes that are factors of 12.

And we can see how our outcome tree compares to our Venn diagram.

So for example, here we have all the outcomes that fall in event A.

So one, four and nine, and we can see them inside the purple circle in the Venn diagram as well.

Here we've got the set of outcomes that fall in event B, so they're factors of 12.

And we can see the numbers 1, 2, 3, 4, 6, and 12.

Here what all the outcomes that were not square numbers and when not factors of 12.

And you can see on the outcome tree we've gone down the branch not A and the branch not B.

And since they're not in either event, they have to go outside the circles, but within the rectangle which represents our sample space.

The nine here, can you see that? We can see that is a square number, but it's not a factor of 12.

So on our outcome tree we went along the branch that said A, so event A is true, but then we went down the branch for not B.

So in other words, we're not a factor of 12.

And that's why the nine is within the circle that represents event A, but not part of the circle that represents event B.

Let's check that.

Use the outcome tree to complete the Venn diagram.

So some of the values have been filled in for you, but you need to fill in the missing ones.

Pause and do this now.

Welcome back.

Did you spot which numbers were missing? Well, luckily we had the numbers one to 12, so it was pretty easy to spot which ones were missing from the Venn diagram, we can see three is missing and we can see six is missing and we can see 12 is missing.

So three, six and 12 need to be filled in.

Let's see where they go.

And they go in the space where A and B overlap three, six, and 12 are all multiples of three and or factors of 12.

So in fact, what was missing were the outcomes from that very top set of branches on our outcome tree.

This Venn diagram represents Alex, Jun, Jacob and Sam, and event A refers to visiting Spain and B refers to visiting France.

We can see from this Venn diagram that three of the four children have visited either Spain or France.

Can you see them? That one of the four has visited both Spain and France.

One of the four has not visited either Spain or France.

How many people have visited Spain? That's right, there's two people who have visited Spain and that would be Jun and Sam.

How many people have visited France? That's right, again, it's two.

We have Jacob and Sam.

And who has visited both France and Spain? That's right, Sam.

We can see him right there in the middle where the two circles overlap.

Now, who has not visited France or Spain? That's right, that's Alex.

Alex is not in either of the two circles and therefore is not part of the set of outcomes for either event.

Using the Venn diagram here, answer the following questions.

So they're gonna appear one at a time.

And the two events are studies French, and that's A and study Spanish, and that's B.

So which pupil studies Spanish only.

So they only study Spanish.

Pause while you write down the name of the pupil.

Welcome back.

Who did you put down? You should have written down that it's Sofia.

She's in the circle that represents event B, but she's not in the part of the circle that overlaps with A, and therefore she only studies Spanish but she does not study French.

Next question, which pupils study French? Pause and write down the names.

Welcome back.

Did you write Andeep and Aisha? Well done if you did.

They are inside the circle that refers to event A.

So which pupil studies neither French nor Spanish? Pause and write it down.

Welcome back.

So who is it? That's right, it's Laura because she's not in either circle.

So she doesn't satisfy either event.

So a Venn diagram is going to be used to represent the sample space of this spinner.

The two events are A factors of three and B multiples of two.

And you can see the Venn diagram here on the right.

How many common outcomes are there between event A and B? So which outcomes are both factors of three and multiples of two? Are you thinking that there are any or have you spotted something? That's right.

There are no common outcomes because the intersection where the two circles overlap is empty.

We could in fact draw a Venn diagram like this.

So it's possible to have a Venn diagram where the two circles do not overlap, and that's because there is no outcome that is common to both events and that's absolutely fine to draw them like that.

So here we've got a Venn diagram being used to represent the sample space of this spinner.

And the two events now are A is even and B is multiples of four.

So question for you, how many of the outcomes are only in event B? So they're only multiples of four.

They are not even.

Again, did you spot all of the outcomes of event B are also outcomes of event A? Because if they're multiples of four, they're automatically even.

So there are no outcomes that are owning event B, and that's why we have an empty area.

So we could have drawn our circles like this.

Which of these Venn diagrams would be correct to use for a trial where I'm rolling a first six excited dice.

And event A refers to the outcome being a multiple of two and event B refers to it being an odd number.

So would that be A, B or C? Pause and make your choice now.

Welcome back.

Which ones did you go for? Well, you could have put A, there's nothing wrong with that because although the circles overlap, I don't have to put anything in that intersection.

I could just leave it blank.

And that would tell me that there isn't an outcome that satisfies both events.

You could, of course have gone for B, because I don't have to draw my two circles overlapping because there is no outcome that satisfies both.

However, C is definitely wrong because there are no multiples of two that are also odd.

Time for our first task, here's an outcome tree for two events.

Please draw a Venn diagram to show the same information.

And I've told you that event A is multiples of three and B is factors of 15.

Pause and do this now.

Welcome back.

Question two.

Here's an outcome table for two events.

Please draw me a Venn diagram to show the same information.

Pause and do this now.

Welcome back.

Question three.

Here's a partially complete outcome table and a partially complete Venn diagram.

So what you're going to need to do is to use one diagram to complete the other and vice versa.

Pause and do this now.

Welcome back.

Time to go through our answers.

Now you could have drawn the two circles A and B round the other way.

So that B was the circle on the left and A was the circle on the right.

But just make sure that the numbers in each area on your diagram match up to mine.

Pause if you want to check your diagram against mine and would like a bit more time.

Question two, here's the Venn diagram, and again, you need the same outcomes in each area on your Venn diagram, even if your Venn diagram is in a different order to mine.

So do pause if you need some more time to check.

Question three you had a partially complete outcome table and a partially complete Venn diagram and I ask you to complete it.

Do feel free to pause to check you filled in that missing information correctly.

It's now time for the second part of today's lesson and that's on outcomes from Venn diagrams. So here's a Venn diagram for the two events P and O where P has the set of outcomes 2, 3, 5, and O has the set of outcomes 1, 3, 5.

Here's the event that an outcome is prime and O is the event that outcome is odd.

Two is only prime 'cause it's not an odd number.

And three and five are both prime and odd, which is why they're in the intersection.

Four is neither prime nor odd, which is why it's outside the circles, but it is inside the sample space.

Quick check.

Event A is referring to factors of 10 and event B refers to even numbers, which outcomes are both in event A and event B.

Pause while you write this down.

So 2 and 10 are the outcomes that satisfy both events.

Which outcomes are neither in event A or event B, pause while you write them down.

The set of outcomes that are neither in event A or in event B are three, seven, and nine.

Which outcomes are either in A or in B? Pause and write this down.

The set of outcomes that are either in A or B are 1, 2, 4, 5, 6, 8, and 10.

Now you might not have wanted to write 2 and 10 because the question said that either in A or B, and you might have thought the or means they had to be in one or the other but not both.

Now we didn't say that we just said it had to be in A or B, so it's absolutely fine for them to be in both and they should be included here.

This Venn diagram has been completed for two events A and B.

And we can see from the Venn diagram there are three outcomes in event A, there are three outcomes in event B.

There is one common outcome in event A and B and there is one outcome that is not in event A or B.

Did you identify all of these? Let's just check then list the outcomes that are in event A.

You should have put two, three, and five.

What's about the outcomes that are in event B? You should have put two, four, and six.

So which outcomes are in both A and B? That's right, it's two.

Because two is in the intersection of the two circles.

Quick check now.

I'd like you please to fill in the blanks.

The Venn diagram shows that there are how many outcomes in event A, how many outcomes in event A and B.

And then lastly, filling in the number for how many outcomes are not in event A or event B? Pause the video while you do this now.

Welcome back.

Let's check our answers.

There are three outcomes in event A.

That's one, three, and five.

There are two outcomes in event A and B, and that's three and five.

And there are two outcomes that are not in event A or event B, and that's four and six.

It's time for our final task now.

Please use the Venn diagram to answer the following questions.

So we've got A, list the outcomes that are in event A.

B, list the outcomes that in event B.

C, list the outcomes that are in both A and B.

D, list the outcomes that are in A, but not in B.

And then E, list the outcomes that are not in either event A nor event B.

Pause the video now while you work this out.

Welcome back.

Question two now.

For part A, use the following information to complete the Venn diagram.

So you've got some clues here and you need to work out what's going where.

And our clues are as follows, the possible outcomes from the trial are the integers from 1 to 10 inclusive.

Second clue there are five possible outcomes for event A.

Third clue six is the only outcome of both events.

Clue four, 2, 5, and 8 are not outcomes of event A nor event B.

And then our final clue is the outcomes that are only in event B are prime.

So use those clues please to complete the Venn diagram.

For part B, now list the outcomes of event A.

And part C list all the outcomes for event A or event B, and we're being really clear here or both.

Pause the video and do this now.

Time now for question three.

Use this Venn diagram to answer the following questions.

Part A, how many outcomes are in the sample space? B, how many outcomes are in event A? C, how many outcomes are in event B? D, how many outcomes are not in event A or event B? And then for part E, can you explain why your answers to parts B, C and D do not sum to make your answer to part A.

Hmm, bit of thought for that last one needed.

Pause the video while you have a go at this now.

Welcome back.

Time to go through our answers.

So question one, the outcomes through in event A are 1, 2, 3, 7, 8, 12, 13, and 15.

In B, the outcomes through in event B are 2, 4, 5, 8, 9, 14, and 15.

The outcomes that are in both events can be seen in the intersection of the two circles, so that's 2, 8, and 15.

Now the outcomes that are in A but not in B are all the outcomes that are inside the circle showing event A but are not in the intersection.

So that's 1, 3, 7, 12, and 13.

And finally part E, the outcomes that are not in either A or B were 6, 10, and 11.

For question two, you had to use the clues to complete the Venn diagram.

So remember we knew firstly that our sample space comprised the numbers of one to 10 inclusive.

So I actually listed that out at the top of the Venn diagram.

So I knew what numbers were going to be going into my Venn diagram.

I knew that five of those numbers were going inside this circle that represents event A six is the only outcome of both.

So six had to go in the intersection and it's the only one that goes there.

So you could at this point have crossed six off from your list at the top to show the outcomes been covered.

Now 2, 5 and 8 are not outcomes of event A nor B.

So they had to go outside the circles, but inside the sample space, so the rectangle, and again I could have crossed off 2, 5 and 8 right now.

At this point I filled in four of the 10 numbers.

It then said the outcomes that are only in event B are prime.

So let's look at what I had left.

Well, I have two regions to fill in and that's the region that represents just event A and it can't overlap with event B.

And I have the region that represents event B, but it cannot represent A as well.

So it's really easy now to do this because if it's prime it's gonna go in B only, and if it's not prime then it's going in A only and that's why 1, 4, 9 and 10, 'cause they're not prime, but they were left over, had to go in just A and 3 and 7 which are prime had to go in just B.

So part B, list the outcomes of event A.

Well there's 1, 4, 6, 9, and 10.

Don't forget the number in the intersection.

It is still part of event A.

And then for C, the outcomes for event A or B or both.

So we had 1, 3, 4, 6, 7, 9, and 10.

Well done if you got this right.

In question three, how many outcomes are there in the sample space? Well, there are 12 values there, which means we have 12 outcomes.

How many outcomes are in event A? And there are six numbers I can see there.

For C, how many outcomes are in event B? Well, there are five.

For D, how many outcomes are not in event A or B? And there are just three that you can see and that's 18, 44 and 62.

For part E, you had to explain why your answers to parts B, C and D.

So that was three, five, and six.

When I sum those together, I get 14.

And that was me covering outcomes in event A, outcomes in event B and outcomes that were in neither, but that didn't sum to make 12, which is how many outcomes are in the sample space altogether.

Why is that? Well, when I counted the outcomes for event A, I counted the 15 and the 20 and when I counted the outcomes for event B, I counted that 15 and 20 again.

So those two outcomes exist in both event A and event B, so they ended up getting counted twice, which is why I ended up with 14 instead of 12.

It's now time to summarise what we've learned today.

Venn diagrams can show the outcomes for two events, outcome tables and outcome trees can be used to generate a Venn diagram and Venn diagrams can be used to generate outcome tables and outcome trees.

So in fact, if we know one, we can generate the others and vice versa.

So you can actually move between all three representations.

That's pretty amazing.

Well done today.

I hope you've really enjoyed learning about Venn diagrams and I can't wait to see you for more of our lessons on probability.