Lesson video

Hi.

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

This lesson falls within the unit "Probability, Possible Outcomes".

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

Let's get started now.

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

So here's a key phrase that we're gonna be using today, and that's the phrase, Venn diagram.

Now, you may be very familiar with what Venn diagrams are, but if you are not, you may wish to pause the video right now to read through our definition or maybe go back and look at the lessons on Venn diagrams. But when you're ready, press play and we'll get started.

Today's lesson has two parts to it, and in part one we're going to be looking at Venn diagrams. When the spinner is spun, the outcomes are 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.

And we refer to this as the sample space.

It is the list of all possible outcomes.

The outcomes can be sorted into a Venn diagram for two events.

So for example, we could consider that an outcome is even, or that an outcome is a factor of 20, and you can see that the intersection, i.

e.

, where the two circles cross that region there where 2, 4 and 10 are, those are even numbers that are also factors of 20.

Now, if a third event, square numbers was added, do you think there's a way to show this on a Venn diagram? Izzy thinks, "Well, maybe the structure of the Venn diagram would look like this.

We just have to put an extra circle in." Does each event have the correct outcomes in it? What do you think? Well done if you spotted that the answer is no.

4 is a square number as well as being an even number and a factor of 20.

It needs to be in all 3 events, and it's only currently in 2.

Well, I can't see a place to put 4 on this diagram so that it's in all three circles.

What are we gonna do, Izzy? Is there a way that we can do this? Ah, well spotted.

Izzy's found a way to draw all 3 events so that there is an area in our Venn diagram where all 3 events can be true.

Can you see that? So the region that the 4 is in, it's inside circle A, inside circle B, and it's inside circle C.

Now this region that's been coloured is the intersection for events B, factor 20 and C, square numbers.

And 4 and 1 are indeed factors of 20 and square numbers.

This area that you can see here is for events that are in B and C but not A.

So we have here factors of 20 that are square numbers and are not even.

What does the shaded section of this Venn diagram represent? So you don't necessarily know what A, B and C stand for, but you could tell me whether this shady section would be true for 1, not true for 1? Think about how you could phrase this.

Pause the video and write down what you think the answer is now.

Pause the video while you have a go at this.

Welcome back.

This shaded section represents outcomes that are in event A and in event C, but they are not in event B.

So you might have said there are outcomes that are true for A, true for C, but not true for B.

Anything like that where you are telling me that it's in A and in C, but not in B would be correct.

What does this shaded section of the Venn diagram represent? Pause the video and write down your answer.

Well, this shaded section represents outcomes that are only in event A.

As you can see, nothing in that area is in B or C.

This Venn diagram represents some pupils.

We have pupils that have visited Spain, that's event A, visited France, event B and visited America, event C.

We can see from the Venn diagram that 8 pupils are represented here, that 1 pupil has not visited Spain, France, or America, and that 4 pupils have visited America, 2 pupils have visited both France and America.

Let's do some questions on this then.

Who has visited Spain? Pause the video and write down your answers.

Right, well, which pupils are in the circle that represents event A, visited Spain? And we can see there are 3.

We have Lucas, Sam, and Jun.

Who has visited Spain, France, and America? Pause the video now and write down your answer.

Right, we need a pupil that appears in all 3 circles.

There's only one and that's Sam.

Sam is in the intersection of events A, B, and C.

Who has visited only France? So they've only been to France.

Pause the video and write down your answer now.

Welcome back.

Remember, they need to be in the circle that represents event B, but they can't be in any other circle.

So that's just Jacob and Aisha.

This Venn diagram shows the menu choices of 8 pupils.

So what you can see here is G represents Garlic mushrooms,, F represents Fish and chips and I represents Ice Cream.

We know the three courses chosen for some of these pupils.

So Alex, for example, has chosen Garlic mushrooms because he's in the circle for G, he's chosen Fish and chips because he's in the circle for F, and he must have chosen Fruit salad because he's not in the circle for ice cream.

Jacob meanwhile must have chosen Halloumi bites because he is not in the circle for Garlic mushrooms, he's chosen Fish and chips because he's in the circle for F, and he must have chosen Fruit salad because he is not in the circle for I.

Sam must have chosen Halloumi bites, Fish and chips and Ice cream.

Can you see that he's in the circles for F and I, but not for G? Now for other students, we dunno all three courses, and this would be for example Izzy.

Izzy has chosen Garlic mushrooms and Ice cream, but she's not in the group of pupils that have chosen Fish and chips.

The problem is there are two options left over to her.

She might have chosen a Bacon cheeseburger, but she might have chosen Vegetable curry and we don't know which one she has chosen.

We just know what she hasn't chosen.

So the Venn diagram here shows the menu choices again of those 8 pupils.

Now the situation's changed.

Laura is coming to the meal.

She orders her Halloumi bites, Vegetable curry and Fruit salad.

Add her name to the Venn diagram.

So where is she going to go? Pause the video and do this now.

Welcome back.

Where have you put Laura? Laura should have been added to the sample space.

So in other words, she's within the square, you can see, but she's not in any of our circles and that's because she hasn't ordered Garlic mushrooms, she hasn't ordered Fish and chips and she hasn't ordered Ice cream.

It's now time for your first task.

For question one, a spinner has the possible outcomes of 3, 5, 6, 8, 10, 12, 15, 18, 20, and 25.

Event A is for odd numbers.

Event B, factors of 30 and event C, multiples of 3.

Draw a Venn diagram please to show this information.

For question two, there are 20 playing cards and each card is either red or black and has a number on it from 1 to 10.

For example, R3 would be a red card with a 3 on it.

Now event A is all the black cards.

Event B considers a card has an even number on it.

And event C is if we have a factor of 10.

Please draw a Venn diagram to show this information.

Pause the video and do this now.

Welcome back.

Question three, match the statements to the shaded areas in the Venn diagram.

So in other words, if you look at the first Venn diagram, you'll see we have quite a large purple area shaded there.

What does that area represent? Choose from one of the statements below and then do the same for the remaining two Venn diagrams. Pause and do this now please.

Welcome back.

Time to go through the answers.

So you had to draw a Venn diagram to show the information and I've got one here for you.

Now where you put A, B and C may vary and that's fine if it does.

The important bit is that your outcomes have to be in the same regions as mine.

So for example, wherever you have event A, 25 must be the only outcome that is only in A and no other event.

So although your diagram may be in a different order, your outcomes should all be in the correct respective regions.

Feel free to pause the video while you check this now.

Let's look at question two.

So here you had to draw the Venn diagram to show the information regarding the playing cards.

And again, you may have drawn these event circles in different places and that's fine if you have.

But remember, your outcomes should still be in the correct respective regions.

So feel free to pause the video now while you check your own work.

Time now to look at question three.

You had to match the statements to the shaded areas.

So for the first diagram, we had outcomes of event B only 'cause that shaded area is not within A or within C.

For the second diagram, you can see here the shaded area is in A, it's also in B, but it is not in C.

And then finally what we have here is the outcomes of event A.

I personally think that was the easiest one, so I would probably have matched that one up first.

Well done if you got these right.

It's now time for the second part of our lesson on outcomes from a Venn diagram.

Here is a Venn diagram for 3 events, A, B, and C, where A is a prime number, B is an odd number and C, a factor of 12.

And you can see I've placed the outcomes into the Venn diagram.

9 is only odd.

It is not a prime number nor a factor of 12, which is why it's been placed in this region.

3 is prime, odd and a factor of 12.

So it's gone in the intersection for all 3 events.

8 and 10 are not prime, odd or factors of 12.

So they're outside all of the circles.

Quick check.

Which outcome is in all 3 of the events you can see here.

Pause and write down your answer.

Welcome back.

Did you pick 10? Well done if you did.

It's in the intersection or the crossover of all 3 events.

Which outcome is only a factor of 10? Pause and write your answer now.

Did you put 1? Well done if you did.

1 is the only outcome that is just in A and no other events.

Which outcomes are both factors of 10 and multiples of 5? Pause and write your answer now.

Welcome back.

So which outcomes are factors of 10 and multiples of 5? So they had to intersect A and C, and that's 5 and 10.

The Venn diagram here has been completed for 3 events, A, B, and C.

We can see from the Venn diagram that there are 7 outcomes in event A.

There are 4 outcomes in event B.

There are 5 outcomes in event C and that there are no outcomes that are not in event A, B, or C.

Time to fill in the blanks please.

The Venn diagram here shows that there are how many outcomes in event A, how many outcomes in event A and B and how many outcomes in event C only? Pause and write this down now.

Welcome back.

You should have said that there are 6 outcomes in event A.

That's 1, 3, 5, 7, 9 and 11.

There are two outcomes in events A and B, and that's 3 and 9 and there are 2 outcomes in events C only, 2 and 10.

This outcome tree has 3 events, A, B, and C, and there are 8 possible outcomes.

We can see them listed here at the end of the tree.

Remember I generated these outcomes by following along the branches.

Now we can find these outcomes on a Venn diagram as well.

By considering where each event is located in the Venn diagram, we can place the possible outcomes.

So we're going to do that now.

So it's in A, in B and in C, and that's where 1 has to go.

For the next, we're still in A and B, but this time not C.

So that's where 2 goes.

Now we're still in A, but this time not in B, but we are in C.

So that's where 3 goes.

But if we're in A, not in B and not in C, that's where the 4 goes.

We are not in A now, but we are in B and we're in C so that's where the 5 goes.

But when we are not in C, that's where the 6 is.

We then go back and say we are not in A, and we're not in B, but we are in C and that's where the 7 goes.

And then if we're not in any of the events, we're outside and that's where the 8 is.

Likewise, we can generate an outcome tree from a Venn diagram.

So we can see the outcomes that are in A, that's 1, 3, 5, 7, 9, 11 and then not A had to be everything else, 2, 4, 6, 8, 10 and 12.

Now looking at the outcomes between A, which are also in B, well that's 3 and 9, and the ones that are not are 1, 5, 7, and 11.

Now considering not A, which ones are in B? Well that would be 6 and 12 and the ones that are not in B is everything else.

So that's 2, 4, 8, and 10.

Now, considering the outcomes that were in A and in B and are in C, well that's just 3 and the 9 is not in C.

If we consider in A and not in B, the ones that are in C are 5 and not in C would be 1, 7 and 11.

Now remember not in A that is in B and in C is just 6 and therefore 12 is not in C and then not in A, not in B, that they are in C is 2 and 10 and then not in any of them must be the 4 and the 8.

Which representation do you prefer? Take a moment to pause the video and maybe discuss this with a partner or with a family member.

Did you discuss? You're gonna get a chance to really think about which representation you prefer in a future lesson, but it's good to take time to think about this sort of thing now.

For this quick check, I'd like you to complete the missing outcomes and you can see there are some missing from the outcome tree and there are some missing in the Venn diagram.

Pause the video and do this now.

Welcome back.

How did you get on? Well, you should have filled in the following.

So from the Venn diagram we had a 7 and a 6 missing, and in the outcome tree, we had 2, 4, 8, 10, missing by not B, and we had a 3 missing from the C at the top.

This Venn diagram represents some pupils.

What we have here is a section of the outcome tree for you to complete.

So by looking at the Venn diagram, you should be able to fill in the three gaps from our outcome tree.

Pause the video and do this now.

Welcome back.

Let's see how you got on.

Well, if I'm considering Izzy, Aisha and Jacob being in B, so the statement that must be true is that they are not in event A 'cause it's not possible for Jacob, Izzy, and Aisha to be in A and then in B.

So we are saying, well, what was true before, could these pupils have been in A, they couldn't have been.

So it has to be not A as the event at the front.

Then we have to consider they're not in A and not in B.

Well that's Laura and Andeep.

And then following those branches along for phase 3, we can see C is Laura, so not C must be Andeep.

Well done if you've got this right.

It's now time for our final task.

For question one, you have to use the Venn diagram to answer the following questions.

Pause the video and take time to answer these now.

Welcome back.

Question two.

The Venn diagram shows which pupils have chosen history, maths or psychology as their GCSE choices.

What'd like you to do is to use the Venn diagram to complete questions A and B and then use it to complete the section of the outcome tree that you can see below.

Pause the video and do this now.

Welcome back.

Let's go through the answers.

So using the Venn diagram, you had to list the outcomes in event A, and we can see that's 2, 4, 6, 8, 10, and 12.

The outcomes that are in events C are 1, 2, 3, 4, 6, and 12.

Then for part C, list the outcomes in both B and C.

Well that's just 2 and 3.

Part D, list the outcomes that are only in B, well that's 5, 7, and 11.

And then E, list the outcome that is not in any of the events.

And that's 9.

Question two.

You had to state which pupils have chosen to study history.

So which names are in the circle with an H denoting it? In other words, which pupils have chosen history? Well that's Alex, Aisha, Jun, Sam, and Sophia.

Part B, which pupils have chosen to study history and psychology? Well the only pupils that have chosen to study history and psychology are Sam and Sophia.

Part C, you had to complete the section of the outcome tree.

Well, we had to look at the information we had been given.

If we're considering the event that pupils have chosen maths, we can see Jun and Sam are there.

Now Jun and Sam are in the intersection between history and maths.

So you could have known that the event that when at the start was they have chosen history.

So students that have chosen history and chosen maths are Jun and Sam and not maths, but they have chosen history would be Alex, Aisha and Sophia.

So from there we know we've chosen history and maths, we are now considering have they picked psychology, and if they have they must be Sam.

Well done if you got this all right.

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

Possible outcomes for more than two events can be displayed on a Venn diagram.

Possible outcomes in a list or an outcome tree can generate a Venn diagram and Venn diagrams can generate a list of possible outcomes or an outcome tree as we saw in our lesson today.

Well done.

You've done a great job today and I look forward to seeing you in the future for more of our lessons on probability.