Lesson video

Hello and thank you for choosing this lesson.

My name is Dr.

Rowlandson and I'm excited to be helping you with your learning today.

Let's get started.

Welcome to today's lesson from the unit of Conditional Probability.

This lesson is called Probabilities in Three Event Venn Diagrams, and by the end of today's lesson, we will be able to use three event Venn diagrams to calculate probabilities including conditional probabilities.

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

The lesson is broken into two learning cycles, and we're going to start by interpreting Venn diagrams. Here we have Izzy and Sam.

They conduct a small survey to find out which GCSE subjects are the most popular ones 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 the Venn diagram we can see on the right-hand side, and we can see that there are some pupils who take multiple of those subjects.

The Venn diagram shows a sample space, which are all the possible outcomes, and they are sorted into three events.

event A is that a pupil takes Art, and they include the pupils, Lucas, Alex, Sofia, Jun, and Izzy.

event D are the pupils who take Drama, and event M are all the pupils who take Music.

If a random pupil from this sample space was selected, what is the probability that they would study Art? Perhaps, pause the video while you think about how we might go about working that out from this Venn diagram, and press play when you're ready to continue together.

Well, we could write this probability down using formal notation, and it'll look something a bit like this.

It represents the probability of event A happening.

The probability of event A happening is 5/10.

The 5 in the numerator comes from the number of outcomes in event A.

They are the pupils, Lucas, Alex, Sofia, Jun, and Izzy.

The 10 in the denominator represents the outcomes in the sample space.

They are all 10 pupils who are included in this survey.

So the probability is 5/10.

So let's check what we've learned there.

Here we have the same Venn diagram.

If a pupil is chosen at random, what is the probability that they study GCSE Drama? Pause the video while you write down your answer and press play when you're ready to see what it is.

Well, the probability that they study Drama could be written as P(D), where that means the probability of event D happening, and it would be 5/10 as well, because there are 5 pupils who study GCSE Drama.

They are Jacob, Alex, Sofia, Sam, and Aisha, and there are 10 pupils altogether in this sample space.

How about if a random pupil is selected, what would be the probability that event M would happen? Pause the video while you write down your answer, and press play when you're ready to see what it is.

Well, the probability of event M happening would be the probability that the pupil who is chosen takes GCSE Music, and that would be 6/10.

There are 6 pupils who are in event M.

They are Jun, Izzy, Sofia, Sam, Aisha, and Laura, and there are 10 pupils in the sample space altogether.

Let's take a look at the last three probabilities that we worked out, and that is based on the idea of if we select a pupil at random from this survey, the probability that they would do Art is 5/10.

The probability they would do Drama is 5/10, and the probability that they would do Music is 6/10.

And Izzy noticed something interesting about this.

She says, "Looking at these probabilities, "event A and event D are equally likely." Sam says, "In our survey, the same number "of pupils took Art as Drama," absolutely.

Sam also says, "Our sample is very small, though.

"A bigger sample may have different outcomes." So while the data from this survey would suggest that events A and D are equally likely, a much larger survey might suggest that they are not equally likely.

Now, Sam notices something else.

Sam says that the probabilities of A, D and M sum to 1.

6, "but I thought the probabilities "should add up to 1 whole." Hmm, why has that happened then? Pause the video while you think about why these probabilities do not sum to 1 whole, and press play when you're ready to continue together.

Probabilities sum to 1 whole when the events are exhaustive and mutually exclusive, but what we can see here in this scenario is the events A, D, and M are not mutually exclusive.

Izzy says, "These events are not mutually exclusive "and that's because a pupil could take "more than one of these subjects." Therefore, the pupils who do more than one subject will be accounted for in more than one of these probabilities.

For example, Alex does Art and Drama, therefore, Alex is counted in the probability for Art and the probability for Drama.

Sofia does all three subjects, so she would be accounted for in all three of those probabilities, and that's why they don't sum to 1 whole.

Let's now use this Venn diagram to work out some slightly more complex probabilities.

For example, if a random pupil is selected, what is the probability that they do not study GCSE Art? We could write down this probability using formal notation, and it would look something a bit like this: P and in brackets we have A and our prime symbol, that means the probability that event A does not happen.

that probability would be 5/10.

The reason why, would be that the numerator is the number of outcomes that are not in event A, and they are Jacob, Sam, Aisha, Laura, and Andeep.

Those are the pupils who do not study Art.

And then, the denominator, 10, are the outcomes that are in the sample space altogether, all 10 of those pupils.

Sam notices that A' is a complement of A, therefore the probabilities of A and A' sum to 1 whole.

The event that A happens and the event that A does not happen are mutually exclusive, they can't both happen at the same time and they're exhaustive because they cover all possible outcomes so either a pupil does Art or they do not do Art therefore, the probabilities of those events sum to 1 whole.

How about if a random pupil is selected, what is the probability that they study Art and Music? Well, we could write this probability down using this formal notation here, which means the probability of events A and M happening.

Well, we have an intersection symbol in between the A and the M.

We can see this intersection very visually in a Venn diagram because it's where the two circles for A and M overlap.

The probability would be 3/10 where 3 is the number of outcomes that are in both event A and M.

They are the pupils, Sofia, Jun, and Izzy.

And the 10 are the number of outcomes in the sample space altogether.

So let's check what we've learned.

Here's a probability for you to work out.

If a pupil is selected at random, what is the probability that they study Art and Drama? Pause the video while you work it out and press play when you're ready for an answer.

We can write down this probability using full notation and it would look like this with an intersection symbol between A and D, and that probability would be 2/10 because there are 2 pupils in that intersection out 10 pupils altogether.

So why is Lucas not included in this probability? Pause the video while you write down an answer and press play when you're ready to see the reason why.

The reason why Lucas is not included in that probability is because the outcome that he is chosen at random does not satisfy both events A and D.

It only satisfies event A.

In other words, he takes Art but not Drama.

Whereas this probability is about the event of someone taking both of those subjects.

So when two events are combined using and, the outcomes have to satisfy both of the events.

Let's take a look now, at some other probabilities we could work out.

If a pupil is selected at random, what is the probability that they study Art, and Music, and Drama? Well, we could write this probability down using this formal notation here, which means the probability of events A, M, and D all happening, and we have an intersection symbol in between each event.

This will be, in the Venn diagram, the point where all three of those circles for A, D, and M overlap, and we look at the number of outcomes in that part of the Venn diagram, we can see the probability will be 1/10 where 1 is the number of outcomes that satisfy all three of those events, and that is the outcome that's Sofia is chosen at random.

The 10 represents all the outcomes in the sample space altogether.

Let's work out another probability.

If a random pupil is selected, what is the probability that they do not study any of these subjects? Well, we could write this down using formal notation and it would look like this where it means the probability of events not A, not M, and not D happening.

Quite literally, we're picking someone from this class and they do not study Art, they do not study Drama, and they do not study Music.

And in our formal notation, we have intersection symbols in between each of these three events.

This probability will be 1/10th because there is one outcome that satisfies all three of these events.

Andeep does not study Art, Drama, or Music, and there are 10 outcomes in the sample space altogether.

So let's check what we've learned.

Izzy and Sam ask 30 pupils in their class if they take GCSE Art, GCSE Drama, or GCSE Music and have recorded the frequencies in a Venn diagram, and that's what we can see on the right side of the screen.

So rather than writing the names individually, they've written the number of pupils who satisfy each of those categories.

Please could you use this Venn diagram to calculate the probabilities that you can see on the screen in the bottom left? Pause the video while you do it and press play when you're ready for answers.

Okay, let's take a look at these probabilities together.

The first one, on the top left, is the probability that an outcome belongs to the intersection of M and D.

That's the probability that both M and D occur, and that would be 8/30.

The one below it is the probability that the outcome does not satisfy the intersection of M and D, and that would be 22/30.

Those last two probabilities sum to 1 whole because they are mutually exclusive and exhaustive.

The next one down is the probability that an outcome belongs to the intersection of not M and not D.

In other words, the probability that M does not happen and D does not happen as well, that would be 11/30.

The next one at the top right is the probability for A, M, and D all happen.

That would be 2/30.

The next one will be the probability that an outcome satisfies the intersection of the complement of A, M, and D.

In other words, the probability for M and D happen, but A does not happen and that would be 6/30.

And the last one is the probability that an outcome satisfies the intersection of not A, M, and not D.

In other words, where M happens, but A and D do not happen.

And that'll be here in the Venn diagram and it'll be 3/30.

Let's now find some more probabilities, which perhaps use some notation we haven't yet used in this lesson.

If a random pupil is selected, what is the probability that they study Drama, or Music, or they could study both? Izzy says, "I find it easiest to go "through each pupil individually.

"So does Lucas take Drama or Music? "No, he doesn't.

"Does Alex take Drama or Music? "Yes, Alex takes Drama.

"Does Jacob take Drama or Music? "Yes, Jacob takes Drama as well.

"Does Jun take Drama or Music? "Yes, Jun takes Music.

"I take Music but not Drama." So yeah, Izzy should still be counted.

"Does Sofia take Drama or Music? "Yes, Sofia takes both of those subjects.

"Does Sam take Drama or Music? "Yes, Sam takes both of those subjects as well.

"Does Aisha take Drama or Music? "Yes, Aisha does.

"And does Laura take Drama or Music? "Yes, she takes Music.

"Does Andeep take Drama or Music? "No, he doesn't take any of those three subjects." So what we are looking for are the outcomes in this highlighted section here.

These are all the pupils who take Music, or Drama, and the probability that one of those is selected is 8/10.

Izzy says, I could also write this using the notation you can see on the screen there.

That statement means the probability of D or M happening, and that would be 0.

8.

She says, "Probabilities can be written as fractions, "decimals, or percentages." And that symbol you can see which looks a bit like a U, that represents the union of the events.

In other words, all the outcomes from D and M are included in this probability.

Okay, so let's check what we've learned.

On the left, you've got four Venn diagrams each with different parts shaded and they are labelled A to D.

And on the right you have four statements which describe events using formal notation and they are labelled E to H.

Could you please match the Venn diagrams with the statements? Pause video while you do it and press play when you're ready for answers.

Here's what you should have, Venn diagram A matches statement H, Venn diagram B matches statement F, Venn diagram C matches statement E, and Venn diagram D matches statement G.

Here's a scenario for you to work with: Izzy and Sam ask 30 pupils in their class if they take GCSE Art, GCSE Drama, or GCSE Music and they record the frequencies in a Venn diagram, which you can see on the right.

Please could you use that Venn diagram to calculate the probabilities that you can see on the screen written with formal notation? Pause video while you do it and press play when you're ready for answers.

Let's go through some answers and let's start with the probability in the top left.

That's the probability that an outcome satisfies the union of A and D.

In other words, it includes all the possible outcomes from either of those two groups, and that will be 21/30.

The next probability is for the event that is the complement of the event we've just been working with.

It has in brackets the union of A and D, and that's followed by the prime symbol, which means the complement of this.

In other words, what's the probability, that the outcome is not from the union of A and D? All these outcomes here, and that'll be 9/30.

We're looking for the outcomes that are not in A and not in D.

You could perhaps think about how you might write that event using slightly different notation if you want to.

The next probability is the probability an outcome is from the union of not A and not D.

That'll be 24/30 'cause either those things can happen.

Either the person is not studying Art, or the person is not studying Drama and there'll be 24 people who satisfy that criteria.

The next probability is that an outcome satisfies the union of A and not D.

In other words, either they can study Art, or they do not study Drama, and that will be these people here and it'll be 23/30.

Okay, it's over to you now for Task A.

This task has three questions and here is question 1.

Pause the video while you do it and press play for question 2.

And here is question 2.

Pause, while you do this and press play for question 3.

Here is question 3, pause while you do this and press play for some answers.

Here are your answers to question 1.

Pause while you check these against your own and press play for more answers.

Here are your answers to question 2.

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

Here are your answers to question 3.

Pause while you check these and press play for the next part of today's lesson.

Well done so far, now let's move on to the next part of this lesson where we're going to look at conditional probabilities from a Venn diagram.

Here, we have the Venn diagram we saw earlier, which is based on the results of a survey about which subjects these 10 pupils take at school: Art, Drama, and Music.

Let's think about how we can get some conditional probabilities out of this Venn diagram.

If a random pupil is selected, what is the probability that they study Drama given that they study Music? Let's think about this and Sam's going to help us.

Sam says, "In our sample, there are 6 pupils "that study Music," they are Sofia, Sam, Aisha, Laura, Izzy, and Jun.

"And 3 of these 6 pupils study Drama." They are Sofia, Sam, and Aisha.

So in this particular case, our denominator is not going to be the number of pupils in the sample space already because we are already given that they study Music.

We are looking for out of those 6 pupils who study Music, what is a probability that 1 of them chosen also studies, Drama, and that will be 3 out of 6, 3/6, or one 1/2.

Sam says, "I can write this using notation "where we have P, and in the brackets we've got D "and we have a vertical line and then the M." The vertical line represents the words given that, so the probability of that event D happens given that we know event M happens, and that will be 3/6 or one 1/2.

Sam also says, "I could write these probabilities "as a decimal, or a percentage." So let's check what we've learned.

You've got some formal notation that describes the probability of two events.

True or false, those two events have the same probability? Is that true or is it false, and explain why.

Pause video while you do that and press play when you're ready to see an answer.

The answer is false and to explain why, let's start by getting a sense of what each notation means.

This notation here is the probability of selecting a pupil who takes Drama given that they take Music.

Whereas, this notation is the probability of selecting a pupil who takes Music given that they take Drama.

So those two events are different.

We can show the probability of the difference by working out what they are.

The probability that they take Drama given that they do Music is 3/6.

Whereas, the probability that they take Music given that they do Drama, is 3/5.

Therefore, those probabilities are different.

Let's now take a look at the results of a slightly different survey.

Izzy and Sam ask 30 pupils in their class.

if they take GCSE Art, GCSE Drama, or GCSE Music and they record the frequencies in the Venn diagram.

They use the Venn diagram to find the probability of A given D.

That means the probability that a pupil selected at random from this group study Art given that we know they study Drama.

So let's work this out together.

What is the probability of A given D? Izzy says, "I've shaded in the region that I need "and I can see that the probability is 6/30, or 20%," hmm.

Sam says, "You've made a mistake, Izzy, "with the denominator in your fraction.

"You found the probability "of the intersection of A and D." In other words, the event that Izzy has found the probability for is if we chose someone at random from the entire survey, what would be the probability that they do both Art and Drama? But that's not quite what the question asks.

The question asks, what's the probability to do Art given that we know they do Drama? So we are not selecting from all 30 people in this survey.

We are only selecting, really, from those who do Drama.

So the probability will have a different denominator.

Izzy says, "Oh I see, I worked out "the probability of events A and D, "instead of event A given D.

"So there are 13 pupils who take Drama.

"So if we select the pupil from this group, "the probability they take Art is 6/13," not 6/30.

So let's check what we learned.

Could you please use this Venn diagram to work out the probability of A given M? Pause the video while you do it and press play for an answer.

The answer is D, there are 14 outcomes that satisfy M.

So if we know M has already happened, we are selecting from those 14, which means the probability will be 5/14.

Okay, it's over to you for Task B.

This task contains two questions and here is question 1.

Pause the video while you do it, and press play when you're ready for question 2.

Here is question 2, pause the video while you do this and press play for some answers.

Okay, let's go through some answers.

As we work through question 1 each time a answer appears the relevant sections of the Venn diagram will be highlighted as well.

Feel free to pause video if you need to inspect the Venn diagram more carefully.

The answer to question a, is 34/83 and the Venn diagram highlights where those numbers come from.

To part b, it is 37/67.

The answer to part c, is 34/57.

The answer to part d, is 14/26.

And the answer to part e, is 55/103.

And here are your answers to question 2.

Pause the video while check this against your own and press play when you're ready for today's summary.

Fantastic work today.

Now, let's summarise what we've learned.

A three event Venn diagram can be used to calculate probabilities.

A three event Venn diagram can be used to calculate the probability of a subset of events occurring as well.

And a three event Venn diagram can be used to calculate conditional probabilities.

Well done today, have a great day.