Lesson video

Hello there.

You made a great choice with today's lesson.

It's gonna be a good one.

My name is Dr.

Rowlandson and I'm gonna be supporting you through it.

Let's get started.

Welcome to today's lesson from the unit of conditional probability.

This lesson is called Constructing three event Venn diagrams, and by the end of today's lesson, we'll be able to do precisely that.

Here are some previous keywords that will be useful during today's lesson, so you might want to pause the video if you need to remind yourself what any of these words mean and press Play when you're ready to continue.

The lesson is broken into two learning cycles, and we're going to start with Completing Venn diagrams. Here we have a spinner.

When the spinner is spun, the outcomes are the integers from 1 to 10, and we call this the sample space.

The outcomes can be sorted into a Venn diagram for different numbers of events.

For example, we could draw a Venn diagram that shows the outcomes for one event and it would look something a bit like this.

In this particular case, we defined Event A as being all the outcomes that are even numbers.

So the outcomes for Event A are written inside the circle.

They are 2, 4, 6, 8, and 10, and all the outcomes that are not in Event A are written outside the circle.

So that's one event.

How about two events? We could draw a Venn diagram for two events and it would look something a bit like this.

In this particular case, we've defined Event A as being all the outcomes that are even numbers and Event B as being all the outcomes that are factors of 20.

And because there are some even numbers that are factors of 20, these two events overlap.

We call that the intersection.

The outcomes 2, 4, and 10 are in the intersection of Events A and B, so that's two events in the Venn diagram.

How about three events? If Event A are all the outcomes that are even numbers, Event B are all the outcomes that are factors of 20 and Event C includes all the square numbers, what would the Venn diagram look like for that? Perhaps pause video while you think about what it would look like and press Play when you're ready to continue together.

Well, this time, we have three events, so we need to draw three circles and we need those circles to overlap with each other in each possible combination.

It looks something a bit like this.

Izzy says, "This structure seems to work.

Each event intersects with each of the other events." If we put our numbers in, it would look something a bit like this now.

She says, "The outcome of four is an even number, a factor 20, and a square number, so it's in the intersection of all three events." How about the numbers that are in this shaded region here? This is the intersection for outcomes in Event A and Event B, or in other words, the even numbers that are also factors of 20.

How about just these numbers here in this shaded region? What do these represent in terms of the intersection of different events? This intersection is for the outcomes that are in Event A and Event B, but are not in Event C.

In other words, these are even numbers that are also factors of 20 but are not square numbers.

So let's check what we've learned.

Here we have a new spinner with different numbers on to what we're working with before.

Event A are all the outcomes that are even numbers, Event B are all the outcomes that are factors of 20, and Event C are all the outcomes that are square numbers.

Could you please use that to complete the Venn diagram for this spinner.

Pause video while you do that and press Play when you're ready to see the answers.

It would look something a bit like this.

You'll notice that there are no numbers in the interception of all three events.

In other words, there are no numbers on that spinner that are even and a factor 20 and a square number.

There are other numbers outside that spinner but not on this spinner.

And you can see there are some other blank spaces in other parts of Venn diagram as well.

Here's another spinner this time.

Could you please do the same again? Pause the video while you do that and press Play when you're ready for answers.

And here's what your answers should look like.

Once again, you should see that the intersection of all three events is empty, and so are some of the other parts of Venn diagram as well.

This time, you've got four copies of the same Venn diagram, but each time, a different part of it is shaded.

And you've got four statements on the right-hand side, labelled e to h.

What does the shaded region of the Venn diagrams represent? Please could you match up the statements to the shaded regions? Pause while you do that and press Play when you're ready for answers.

Let's take a look at some answers.

Venn diagram A matches statement F, and that is, it shows all the outcomes that are only in Event B.

Venn diagram B matches statement G.

It shows the outcomes that are in Event B and Event C, but are not in Event A.

Venn diagram C matches statement H.

It shows the outcomes that are in Event A and Event B and Event C.

And Venn diagram D matches Statement B.

It shows the outcomes that are in Event B.

Let's now take a look at a different scenario.

Here we have Izzy and Sam who conduct a small survey to find out which GCSE subjects are popular in their school.

They ask some pupils in their class if they take GCSE Art, GCSE Drama or GCSE Music and they record the outcomes in a Venn diagram and that's what we can see on the right.

Izzy looks this and she makes a few observations.

She says, "It looks like the events are all equally likely, as the regions have the same number of outcomes in each.

There are four people who do Art, four people who do Drama, and for people who do Music.

She also says, "There's an equal chance that someone from the sample is taking Art and Drama.

These subjects, therefore, are equally popular.

There is a higher chance, however, that someone takes two or more of these subjects than just one of them, and we can see that there are three people in this survey who do one subject.

They are Lucas, Jacob, and Laura, but there are four people who do more than one subject.

They are Alex, Sofia, Aisha, and Jun.

So it seems that Izzy has come to the conclusion that people are equally likely to take Art as they are to take Drama as they are to take Music, and she's come to that conclusion based on this particular survey.

But can you see a problem with how Izzy has reached this conclusion? Hmm.

Sam says, "Our sample size is very small.

There are only eight people included in this survey, so it may not be representative.

Let's ask some more people." And so they do.

Izzy and Sam ask 30 pupils in their class if they take GCSE art, GCSE Drama or GCSE Music.

Now, this time, they're not gonna have space to write down the names of all 30 people in their Venn diagram.

So instead, they're going to record the frequencies in the Venn diagram.

Here's what they find.

14 pupils take Art.

13 pupils take Drama, 15 take Music, and five pupils don't take any of these three subjects.

So they fill in the Venn diagram and they look something a bit like this.

Hmm, but Sam's noticed a problem here.

Sam says, "I've recorded these frequencies in a Venn diagram, but it doesn't make sense, as the numbers add up to more than 30." Can you think about why these numbers sum to more than 30? And what we might need to do to rectify this? Pause video while you think about it and press Play when you're ready to continue.

Well, one problem here is that there may be some pupils who do more than one subject and they will be counted more than once in this Venn diagram.

For example, there may be some pupils who take Art and Drama.

Those pupils will be counted in the 14 for Art and again in the 13 for Drama, which is why these numbers add up to more than 30.

So what could we do differently? Izzy says, "We need more information about how many people take more than one of these subjects, so we can complete the intersections of the Venn diagram." So they get some more information.

And it looks a bit like this.

Let's now work to complete the Venn diagram based on information we have here.

Now, often, when filling in a Venn diagram with information like this, it can be helpful to start with the intersection in the very middle first and work outwards in the Venn diagram.

That way, if there are any people who are counted more than once in the Venn diagram, we account for them first and we can use subtraction later, so that we don't count them twice.

You'll see how it works as we go through.

Some plans to do precisely that.

It'll be easier to fill the intersections in first.

So let's start at the bottom of the information.

There are two pupils who take all three subjects.

So that number will go in the intersection of all three groups.

There are three pupils who take only Music and Art, so that three would go in the intersection of Music and Art in the space that is not overlapped by Drama.

There are six pupils who take only Drama and Music, so we can fill that in a similar way.

And there are four pupils who only take Art and Drama.

There are five pupils who don't take any of these subjects.

So that number would go outside all the circles, and now we get to the information where we know that some people are counted more than once.

For example, when it comes to Art, there are 14 pupils who take Art in total, but some of those pupils have already been represented in the Venn diagram and they are in the intersections.

There are four, two and three people who take Art and have already been represented.

So the frequency that we are missing for Art are those that only take Art without taking either of the other two subjects.

We can get that by doing 14, subtract four, subtract two, subtract three, and that will give five.

And now, all the numbers that are inside the circle for Art sum to 14, and then we can do the same for Drama.

There are 13 pupils in total who take Drama.

Some of them they account for are already in intersections.

They are the four, the two, and the six.

So we could do 13, subtract four, subtract two, subtract six to get one remaining person who does Drama without doing any of the other subjects.

There are 15 pupils who take Music in total.

Three of them, two of them, and six of them have already been accounted for in the intersections.

So that means there must be four remaining who only do Music.

And now this Venn diagram is complete, we can double check we haven't made any errors by adding up all the numbers we can see in it and checking that they sum to 30, which they do.

And Sam says, "The Venn diagram now makes sense as the numbers do add up to 30." And you may remember earlier that Izzy thought, all the events were equally likely to happen.

Now she's looking at this, she thinks, "It looks like the events are not equally likely, as the regions have different frequencies in each.

There is not an equal chance that someone from the sample is taking Art and Drama.

There is still a higher chance that someone in our class takes two or more of these subjects than just one though." And Sam says, "I agree.

the Venn diagram shows all this." Let's check what we've learned.

Sam conducts another survey and collects the information that you can see on the left-hand side of the screen.

Sam completes a Venn diagram for that information, which you can see on the right side of the screen, but it's not quite right.

Please explain how you know that Sam has made a mistake.

Pause the video while you write something down and press Play when you're ready to continue.

We know Sam has made a mistake because the frequencies do not sum to 100.

Hmm, but the Venn diagram isn't completely wrong.

Some of it is correct.

Which regions did Sam get right on this Venn diagram? Pause the video the while you write something down and press Play when you're ready for an answer.

The regions which are correct are these ones here, the intersections and the frequency that is outside all three events.

So it looks like the errors are with these frequencies here.

Let's focus for a second just on this highlighted region here for Business.

If we know about 30 pupils take business and we know that the numbers in the intersections are correct, could you please correct this region of the Venn diagram? Pause video while you do it and press Play when you're ready for an answer.

Well, the frequencies in this circle should sum to 30, so we could do 30, subtract the other three frequencies that are in that as intersections, and that would give 12, so that must be 12.

How about if we focus now on the new highlighted circle, which represents Geography.

Given that there are 46 pupils who take Geography, could you correct this region of the Venn diagram? We can use a very similar method.

The frequencies in this circle should sum to 46.

We know three of them are correct, so we can do 46, subtract the three that we know to get 24.

And finally let's now focus on the History event.

Could you please do the same again given that there are 47 pupils who take History? Pause video while you do it and press Play for an answer.

We can do the same process again by subtracting the three frequencies and the intersections from 47 to get 23.

And now, when we double check this, we can see that all the frequencies sum to 100.

Okay, it's over to you then for Task A and here is Question 1.

Pause video while you do it and press Play for Question 2.

Here is Question 2.

Pause video while you do this and press Play for Question 3.

Here is Question 3.

Pause video while you do this.

Okay, let's look at some answers.

For Question 1, Part A, your Venn diagram should look at something like this.

Pause while you check against yours and press Play to continue.

And then for Part B, you have to explain why some regions of the Venn diagram will not have any outcomes written in them.

Well, the only even prime number is two, which is not a square number.

Therefore, the intersection of Events A, B, and C will be empty and the intersections of Events B and C will also be empty as a square number cannot also be prime.

Then in Question 2, you had to match the statements to the shaded regions.

The first shaded region shows the outcomes of Event C only.

The second one shows you outcomes of Event A and C, but not Event B.

And the third one shows all the outcomes of Event C.

And then with Question 3, this is what your answer should look like.

Pause the video while you check this against your own.

Well done so far.

Let's now move on to the next part of this lesson, which is looking at interpreting Venn diagrams. Here we have Izzy and Sam who have asked 30 pupils in their class if they take GCSE Art, GCSE Drama, or GCSE Music and have recorded the frequencies in the Venn diagram we can see on the right of the screen.

They're now going to try and interpret this Venn diagram.

Before you start, Sam asks, "Could you please remind me what the psi symbol means?" Izzy helps and says, "The psi symbol means the universal set.

It is the total frequency of our sample." So that means psi would be equal to 30 in this case.

There are 30 people in this survey altogether.

So now they've got that sorted, they start picking out some other numbers.

Izzy says, "How many pupils take Art?" Hmm.

Sam says, "There are 14 pupils who take Art, and I can write this as A = 14." So Izzy says, "How many pupils do not take Art?" Well, Sam says, "There are 16 pupils who do not take Art.

I can write this as A prime equals 16.

Prime is little dash symbol you can see next to the A.

Now, A prime is the complement of Event A.

So in other words, includes all the outcomes but are not in Event A, and they are 16.

Next, Sam asks, "How many pupils take Drama and Music? Hmm.

Well, Izzy says, "There are eight pupils who take Drama and Music.

I can write this as the intersection of D and M is equal to eight." Let's take a look at that notation a little bit more.

That notation is the intersection of Events D and M.

In the Venn diagram, it's where those two circles overlap.

In the context, they are all the pupil who take Music and Drama, and there are eight of them.

Sam says, "Oh, I see.

So that means the intersection of A, D, and M is two.

That's the part of the Venn diagram where all three circles overlap, and they are all the people who take all three of the subjects.

Izzy says, "Yes.

The intersection of A, D, and M is the intersection of those events, A, D and M; Art, Drama, and Music.

There are two pupils who take Art, Drama, and Music." Now, Sam is starting to get into this, but Sam asks, "How would I describe this region here?" Hmm, look at the Venn diagram.

That region is a section where the circles for D and M overlap, but do not overlap with the circle for A.

In other words, we want the frequencies that belong to D and M but do not belong to A.

Izzy says, "We could describe this as the intersection of not A, D and, M, and that will be equal to six.

This is the intersection of Events D and M and the complement of Event A.

In other words, all the things that are in D and M but not in A.

As this is equal to six, it means that there are six pupils who take Drama and Music, but do not take Art.

So let's check what we've learned.

On the left, we've got four identical Venn diagrams, but each time, a different region is shaded.

On the right, you've got four statements expressed with formal notation.

They're labels from e to h.

What does each shaded region of the Venn diagrams represent? Please match up the statements to the shaded region.

Pause while you do it and press Play for some answers.

Let's go through some answers.

Venn diagram A matches statement H.

It shows the regions that are only in Event B.

In other words, it's the intersection of Event B with not A and not C.

Venn diagram B matches statement G.

It is the intersection of Event A and Event C, but not Event B.

Venn diagram C matches statement F.

It is the intersection of Event A and Event B and Event C and Venn diagram D matches statement E.

It is the intersection of Events A and B.

Let's now take a look at another scenario.

Aisha, Sophia, and Laura are on the school running team.

Izzy has drawn a Venn diagram to record how many races each pupil has run, and Izzy says, "This structure seems to work.

Each event intersects with each of the events." Sam says, "Why do you need the intersections for this Venn diagram?" Izzy says that intersections can be used to record when more than one pupil wins.

But can you see a problem with this? Hmm.

Let's see what happens when we fill it in.

We have these frequencies here, and what you'll notice is that there are quite a few blanks in that Venn diagram.

In fact, every intersection is blank.

Sam says, "Why are the intersections in this Venn diagram blank?" Hmm.

Izzy says, "It means there were no draws of first place in the races." Therefore, each race only has one winner, and that means, in this case, the events are mutually exclusive.

Sam says, "This means a slightly different shape of Venn diagram may be more appropriate." Maybe something a bit more like this.

When two or more events are mutually exclusive, it means they cannot happen at the same time.

Therefore, the frequency of those happening at the same time will always be zero.

Mutually exclusive events can be represented in Venn diagrams in a couple of different ways.

One way could be like we saw earlier, where the intersections between mutually exclusive events have a frequency of zero or are left empty.

Another way could be like we've seen here, using non intersecting circles for events that are mutually exclusive.

Here's a different scenario now.

We have another spinner, and when this spinner is spun, the outcomes are the integers from 71 to 80.

The outcomes can be sorted into a Venn diagram that Events A, B, and C and look like the Venn diagram on the right.

Now, once again, you'll notice, there are quite a few empty regions in this Venn diagram.

In particular, you'll notice that every outcome which belongs to B also belongs to A and every outcome it belongs to C do not belong to either outcomes A or B.

So we can represent it like this with this Venn diagram.

But Izzy says, "As there are lots of blank regions, maybe a different shape of Venn diagram can be used." For example, it could look something a bit like this.

It shows the same information as what we just had.

All the outcomes which belong to Event B also belong to Event A.

But there are some outcomes in Event A that don't belong to Event B and all the outcomes that belong to Event C belong to neither Events A or B.

Izzy says, "Event B is a subset of Event A and Events A and C are mutually exclusive." Okay, it's over to you now for Task B.

And here is Question 1.

Pause video while you do it and press Play for Question 2.

Here is Question 2.

Pause while you do it and press Play for Question 3.

Here is Question 3.

Pause while you do this and press Play for Question 4.

And here is Question 4.

Pause while you do it.

Okay, here are the answers to Question 1.

Pause while you check and press play to continue.

Here are the answers to Question 2.

Pause while you check and press Play for more answers.

Here are the answers to Question 3.

Pause while you check these and press Play for more answers.

Here are the answers to Question 4.

Pause while you check.

Fantastic work today.

Now let's summarise what we've learned.

Venn diagrams may show probabilities, frequencies, or individual outcomes.

A three event Venn diagram is easier to complete if information from the intersecting events is completed first.

A three event Venn diagram may have some or all events being mutually exclusive.

A three event Venn diagram may have events which are subsets of other events as well.

Well done today, have a great day.