Lesson video

Hello there and welcome to today's lesson.

My name is Dr.

Rollinson and I'll be guiding you through it.

Let's get started.

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

This lesson is called conditional probability in a Venn diagram, and by the end of today's lesson, we will be able to calculate a conditional probability from a Venn diagram.

Here are some previous keywords that will 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.

Then press play when you're ready to continue.

The lesson is broken into two learning cycles and we're going to start with Venn diagrams and conditional probability.

Here we have a box that contains four types of marbles.

Some of the marbles are blue with a swirl pattern and some are blue without a swirl pattern.

Some are green with a swirl and some are green without a swirl.

And here we have Aisha.

Aisha says, we can represent the frequency of each type of marble in this two-way table.

The columns of this two-way table show us how many marbles contain a swirl pattern and how many do not contain a swirl pattern.

The rows of the table show us how many of the marbles are blue, and how many are green, and a two-way table can be a great way to sort frequency data for a scenario such as this one, but it's not the only way.

Another way could be to represent it as a Venn diagram, and it could look something a bit like this.

Venn diagrams are a great way of sorting a sample space, population, or results from an experiment into groups or subsets based on particular events.

In this Venn diagram, the circle on the left that is labelled S, shows us how many marbles contain a swirl pattern with 18 of those marbles also being blue and 50 of those marbles not being blue.

The circle on the right, which is labelled B, that shows us how many marbles are blue.

18 of those also contain a swirl, and 11 of them do not contain a swirl.

And there's also a number on the outside of both the circles, the number 16, that shows us how many marbles are not swirl patterned and are not blue.

In other words, the green plain marbles.

Venn diagrams are also a useful way to model and answer probability questions.

So let's take a look at some of those now.

Here are some probability questions for you to think about yourself.

Pause the video while you work through these and press play when you're ready to continue.

Let's take a look at some answers.

In the first question, you are asked how many marbles there are in total in the box.

That'll be 60.

It's the sum of all four numbers in that Venn diagram.

The second question asks you how many of the marbles are blue? There are 29 blue marbles in the box.

That's a sum of the two numbers inside the circle that is labelled B for blue.

And then you have three probabilities to work out based on a scenario or drawing a marble at random from the box.

The probability the marble is blue would be 29/60.

That's because there are 29 blue marbles in the box and you could choose any of the 60 marbles that are in the box altogether.

The probability that the marble is a blue swirl marble would be 18/60.

That's because there are 18 blue swirl marbles in the box and you could choose any of the 60 marbles that's in the box altogether and the probability that it's not a green swirl marble, that would be 45/60.

You can either get it by adding together all the marbles that fit that description, all the ones that are not a green swirl marble, so it could be a green plain marble, or it could be a blue swirl marble, or a blue plain marble.

Add those together and you get 45, and then you got 60 because you could choose any of the 60 marbles in the box.

Another way could be to look at how many of the marbles are green swirl marbles and subtract that from 60.

60 subtract 50 is also 45, so it'd be 45 over 60 for that reason.

Let's take a look at one of those probabilities in particular and that is if a marble is randomly chosen from the box, the probability that it'll be a blue swirl is 18/60.

We could represent that with some formal notation.

It'll look something a bit like this.

This part here with the P and the brackets that tells us the probability of events B and S both happening.

The symbol inside the brackets is the intersection symbol, so that means we are looking for the probability that the outcome satisfies the intersection of B and S.

We can see that intersection very visually in the Venn diagram.

It's a bit where the two circles overlap, the bit with the S circle and the B circle both overlap.

That shows us how many marbles satisfy both of those events.

The probability comes in two parts.

The numerator tells us how many outcomes are in that intersection or how many outcomes are in events B and S together, and that is 18.

The denominator tells us how many outcomes are in the sample space altogether, and that is 60.

Aisha says, I could write this probability in a few different ways.

One way she could phrase it could be the probability of blue swirl or she could use a notation, the probability of the intersection of B and S.

In either case, it'll be 18/60.

She also says, I could write the probability as a decimal or a percentage, and that could be as 0.

3 or 30% or she could use any equivalent fraction as well.

Let's change the scenario here ever so slightly because the probability of a particular marble being selected can change if certain conditions have already been met.

For example, a marble is randomly chosen, but before the marble is taken all the way out of the box, part of the marble is visible, and we can tell for certain now that it's a blue marble.

However, we could not yet tell whether or not the marble has a swirl or not.

So if we know it's a blue marble, it means that the probability that's a blue swirl marble is no longer 18/60.

Can you think why? And what is the probability that this is a blue swirl marble given that we already know it's blue? Pause the video while you think about why the probability would change in the situation and what it could be.

And press play when you're ready to continue together.

Well, with the previous probabilities we worked out from this Venn diagram, the denominator was always 60.

That's because the marble we were choosing could have been any one of the 60 marbles in the box, but this time, it can be any one of the 60 marbles because we know it's already blue.

Therefore, it can only be one of the 29 blue marbles that are in the box.

So since we know it's blue, it is guaranteed to be one of these 29 blue marbles.

That means the denominator of the fraction will not be 60 because it can't be any of the 60 marbles.

It'll be 29 because it's one of the 29 blue marbles.

There are 18 out these 29 blue marbles which have a swirl.

Therefore, the probability that this is a marble, which is a blue swirl marble, given that it's already blue, would be 18/29.

This is something that's called conditional probability.

The probability that this marble is a blue swirl marble in this case has changed since we know that it's definitely blue.

Therefore, we could say the probability that it's a swirl marble given that it's a blue marble is 18/29, and this can be written with some formal notation as well.

The word "given" in conditional probability can be written with a vertical line symbol and the vertical line means the words given that something else is already known.

For example, we could write it like this.

P for probability and the brackets swirl, we've got a vertical line and blue, and what that means is the probability that it's a swirl marble given that it's a blue marble is 18/29.

Aisha says, I'm confused.

So the probability of choosing a blue swirled marble could be 18/60 or 18/29.

Is that right? I don't think that is quite right.

The conditional probability of this being a blue swirl marble is different to what the probability would be if we didn't know anything about the marble at all.

Let's illustrate that with two diagrams. On the left, we have a box where we don't know anything about the marble that we are selecting whatsoever.

We don't know whether it's blue, we don't know whether it has a swirl or not.

Therefore, the marble that we are choosing could be any of the 60 marbles that are in that box.

18 of them are blue swirl ones, so probability would be 18/60.

With the box on the right, we can already see that it's blue, therefore, it can't be any of the 60 marbles in the box altogether.

It can only be one of the 29 blue marbles that are in the box.

Therefore, in this case, the probability that it's a blue swirl marble is 18/29.

Aisha says, oh, I understand now.

There are still 18 blue swirl marbles, so the numerator stays the same, but as I now have given information, the denominator changes as the sample space has changed.

That's absolutely right.

In both of those probabilities, the numerator is 18, but the denominator is different because the number of possible marbles we are drawing from has changed.

On the left it's any of the 60 marbles in the box.

On the right, it's any of the 29 blue marbles.

Now you can't blame Aisha for being a bit confused by this to begin with because when you look at the probabilities, the events inside the brackets are worded in the exact same way.

They both say blue swirl, so it's a little bit ambiguous about what these two probabilities are talking about.

Using clearer language and notation will help avoid any confusion and will also help us to see if we are using conditional probability or not.

For example, these probabilities could be rephrased like this.

On the left, the probability it's a swirl marble and a blue marble is 18/60, whereas on the right we have the probability it's a swirl marble given that it's a blue marble and that would be 18/29.

Those two probabilities are now phrased slightly differently to give us a little bit more clarity about what they mean.

The difference is the word "and" and "given." Those could also be written with some notation as well.

We could write it like this.

On the left we are saying, what's the probability that the outcome is from the intersection of swirl and blue? Whereas on the right, we are saying What's the probability of the outcome is a swirl, given that we know it's a blue? And those two are different.

So let's check what we've learned.

A marble is chosen at random from this box.

The Venn diagram shows all the frequencies about the marbles in the box.

We can see the top of the marble shows it has a swirl, but we cannot yet see the colour of the marble.

What is the probability that this is a blue swirl marble, given that we know it is swirled? Pause the video while you write it down.

And press play when you're ready for an answer.

The answer is 18/33.

We are looking for the probability that it's a blue marble given that we know it's a swirl marble and that's what's represented in the form notation on the left.

The fraction is in two parts.

The 18 shows us the number of blue swirled marbles are in the box altogether and the denominator 33 shows us the number of swirled marbles in the box.

So there are 18 out of a possible 33 swirled marbles that could be blue.

Here's a true or false question.

You've got two probabilities written with form notation and a statement claims that those two probabilities are equal to each other.

Is that true or is it false? And explain your reasoning as well.

Pause the video while you write down your answer and press play when you're ready to see what the answer is.

The answer is false.

There are a few different ways you can explain this.

One could be to explain what each notation means.

This notation is the probability of selecting a marble with a swirl given that we know it's blue, and this notation means the probability of selecting a blue marble, given that it's swirled.

Those two different bits of notation mean different probabilities or different events.

We could also be more convincing by working out what those probabilities are.

The probability it's a swirl marble, given that it's a blue marble is 18/29.

The probability that's a blue marble given that it's a swirl marble is 18 over 33.

Those two probabilities are different.

So here we have a different box that also contains blue and green marbles, each of which can either have a swirl or not have a swirl.

A marble is chosen at random.

Could you please work out the probability it's a swirl marble and a green marble? Pause the video while you write your answer down and press play when you are ready to see what the answer is.

The answer is 25 over 140.

We are looking for a marble that satisfies the event.

Where is the intersection of swirl and not blue? There are 25 of those out of a possible 140 marbles in the box.

Could you please now think about this scenario where the top of the marble is green, so you'd know it's definitely a green marble? What is now the probability it's a swirl marble, given that it's a green marble? Pause the video while you write down your answer and press play when you're ready to see what the answer is.

We are looking for the probability it's a swirl marble given that it is not a blue marble and that would be 25 over 72.

The numerator is the same as a previous probability, but what it's possibly out of has changed because we are only looking for the 25 marbles that are out of the possible 72 marbles, which are not blue, and that 72 comes from the sum of 25 and 47.

Those are the ones which are outside the circle labelled B.

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

This task contains three questions, and here is question one.

Pause the video while you do this and press play when you're ready for questions two and three.

And here are questions two and three.

Pause the video while you do these and press play when you're ready for some answers.

Okay, let's go through some answers.

Here are the answers to question one.

Pause the video while check this against your own and press play when you're ready for more answers.

And here are the answers to question two.

Pause the video while you check these and press play when you're ready for the answers to question three.

And here are the answers to question three.

Pause the video while you check these against your own and press play when you're ready for the next part of today's lesson.

Well done so far, let's now move on to the next part of this lesson where we're going to look at mutually exclusive and independent events.

Let's go back to our box of marbles where there are four different types of marbles.

Some are blue with a swirl, some are blue without a swirl, some are green with a swirl, and some are green without a swirl.

The information about the frequency of each type of marble can be represented in a frequency table like one you can see on a screen here or in a Venn diagram like this one here.

Let's take the Venn diagram for a second.

Sophia says, I want to work out some probabilities about the green swirled marbles.

How can I do this using a Venn diagram? Aisha says, you still have the same information on this Venn diagram as in the frequency table.

Can you see why? Perhaps pause the video and look at this Venn diagram and think about how each piece of information that would be in a frequency table is represented somewhere in this Venn diagram, and then press play when you're ready to continue together.

Each piece of information that would be in the two-way frequency table is in this Venn diagram somewhere.

For example, you have the frequency of each individual type of marble represented as those four numbers.

We could also get the totals as well by adding together numbers.

For example, you can get a total number of swirl marbles by adding together 15 and 18 and so on.

So all the information is in there somewhere.

It's just presented in a slightly different way.

Sophia says, I can see that blue marbles go in the circle marked event B, that's the 18 and 11.

She says there are 29 marbles altogether that are blue, 18 of which are swirled.

And Aisha says the green marbles are placed in the not blue region.

In this example, green is the complement of blue, or you can write that as B with the prime symbol.

And once again, you can see there are two different types of green marbles represented.

15 are green with a swirl, and 16 are green without a swirl.

Sophia says, I now see how I can work out the probability of selecting a green swirled marble.

She says the probability of it being a swirl marble and a green marble, which can be written as the probability of the intersection of S and not B, that is 15/60.

There are 15 green swirl marbles out of a possible 60 marbles altogether.

And Aisha says, I agree.

This is the probability if any random marble is selected out of the 60 marbles in total.

But what about a conditional probability? Sophia says, if I already know that the marble is green, I can work out the conditional probability of selecting a green swirl marble.

That would be by doing 15 over 31.

It's a probability of swirl given green, which you can write with S, our vertical line, and the complement of B.

The 15 comes from the 15 green swirl marbles, and the 31 comes from the sum of all the green marbles.

15 plus 18 is 31.

And Aisha says, I agree.

This is because you are only selecting from one of the 31 green marbles.

In this case not selecting from all of the 60 marbles.

Sophia says, I understand how to use Venn diagrams now to find probabilities, but does it always have to be set up in this way? I think by this, Sophia is referring to the events that are shown in the Venn diagram.

S for ones being swirled and B for ones being blue.

Aisha says no.

We could have drawn the diagram in a different way to represent the same information.

For example, we could have blue and no swirl as the two events.

That would look something a bit like this.

The circle on the right shows us how many are blue, and the circle on the left shows us how many do not have a swirl.

You'll notice that the numbers in the Venn diagram are all the same.

They're just in different places on the Venn diagram.

So let's now look at other ways we could represent the same information in a Venn diagram.

There are six different Venn diagrams that can be drawn from the information that is shown in the two-way frequency table on the left.

Two of the Venn diagrams have already been completed.

Could you please complete the other four Venn diagrams to represent the information from that two-way frequency table? Pause the video while you do this and press play when you're ready to see some answers.

Okay, let's now fill in the rest of these Venn diagrams and let's start with the one labelled S and G.

The intersection of those two events would be marbles which are swirled and green, and there are 15 of those.

The remaining marbles in that set for S, they are the ones which are swirled but are not green, and there are 18 of those.

The remaining marbles that are in the group for G but are not in the intersection.

They are the ones that are green but are not swirled and there are 16 of those and the number that go outside both of those circles are the ones which are not green and not swirled.

In other words, the blue no swirled ones and there are 11 of those.

And you can fill in the other three Venn diagrams using a similar method.

Now, you may have noticed something a little bit interesting or unusual about some of the answers you got when you were completing those Venn diagrams, particularly for those two bottom Venn diagrams. Let's take a look at those now.

'Cause Aisha, she spots something interesting.

She said in some of the Venn diagrams, there are zeros in some of the regions.

Sophia says, this is because the events are mutually exclusive in these two Venn diagrams. This means that only one or the other of these events can happen.

Both of the events cannot happen at the same time.

Aisha says that makes sense as we don't have any marbles that are blue and green at the same time.

So with that Venn diagram in the bottom left there we can see the intersection is zero because that means those are the marbles which are blue and green.

There are none of those.

With the Venn diagram on the right, swirl and no swirl, those two things cannot both happen at the same time.

So the number in the intersection is zero in that case as well.

Sophia says the probability of getting a marble which is blue and green or probability for the intersection of B and G occurs is zero out of 60.

Let's take a look at this probability in a little bit more detail now.

If a marble is randomly chosen from the box, the probability that would be blue and green is zero out of 60.

That could be written formally with notation like this is the intersection of B and G occurring.

That probability is 0/60 where the numerator zero shows us how many outcomes are in event B and event G.

That's the intersection of B and G and there are zero of those.

The dominator tells us how many outcomes are in the sample space altogether.

That's a sum of the four numbers in the sample space.

The probability is zero as it is impossible for mutually exclusive events to occur at the same time, and that's what it means for two events to be mutually exclusive.

It is impossible from both to occur at the same time.

So if two events are mutually exclusive, it means the probability of an outcome from the intersection of those two events, will always be equal to zero, and that's how we can tell if they are mutually exclusive.

We can also determine mutual exclusivity by considering conditional probability.

The probability that this is a green marble, given that we already know it's a blue marble, is zero out of 29.

So if we know the marble is blue, then it's impossible for it to be green.

Alternatively, if we know that the marble is green, the probability that it's blue, given that it's green, is zero over 31.

If we know a marble is green, it cannot be blue.

So each of these conditional probabilities are zero as it is impossible for mutually exclusive events to both occur at the same time.

So let's check what we've learned.

If event A and event B are mutually exclusive events, which of these statements are correct? There may be more than one.

Pause the video while you choose and press play when you're ready to see some answers.

Okay, let's now go through these one at a time.

For the statement in A, the probability that A and B both occur is zero.

That would be true because mutually exclusive events cannot both happen at the same time.

For statement B, the probability that A happens given what we know B has happened, is equal to zero.

That is also true because if B has happened, then A cannot happen.

For statement C, the probability that either B happens or A happens is equal to zero.

That statement is false.

One of those events could happen just so long as the other one doesn't happen, so the probability is not zero 'cause it's not impossible for at least one of those events to happen.

And statement D, the probability of that B happens given that A has happened is equal to zero, that'd be true.

If A has happened, then it's impossible for B to also happen.

Here we have a different box now that contains blue and pink marbles, each of which can have a swirl or not have a swirl, and the frequencies of each type of marble is shown in the Venn diagram.

Aisha says, I can see that events S and B are not mutually exclusive because the intersection of S and B is not zero.

We can see the intersection is two, and Sophia clarifies this by saying the probability of a swirled and a blue marble being chosen from the box or probability that an outcome is from the intersection of S and B, is 2/60.

Let's now think about conditional probabilities in this situation.

If we know that the marble is pink, we can find the conditional probability of it also being swirled.

The probability that it's a swirl marble given that it's a pink marble, that would be the probability that it's a swirl marble, given that it is not a blue marble, which can be written formally with notation like this, the probability of S given that not blue.

When we look at the Venn diagram, we can see that there are eight marbles, which are swirl marbles, out of a possible 48 marbles altogether, which are not blue.

So the probability would be 8 over 48.

Let's compare these two probabilities now.

The probability that's a swirl marble given it's a blue marble, is 2/12, and the probability it's a swirl marble, given that it's not a blue marble, is 8/48.

Do you notice anything interesting about those two fractions? Pause the video while you think about what these two fractions may have in common and press play when you're ready to continue.

Well, Aisha has noticed something about these fractions.

She says they are equivalent.

The probability it's a swirl marble given it's a blue marble, is the same as a probability, it's a swirl marble given that it's not a blue marble because 2/12 is equal to 8/48.

They're both equal to 1/6.

She then goes on to say, this is also equivalent to choosing a swirl marble in general, which is 10/60.

There are 10 swirl marbles in total in the box, and there are 60 marbles altogether, and 10/60 is also equivalent to one sixth.

So it looks like the probability of getting a swirl marble is the same regardless of whether it's blue, not blue, or just a marble in general from the box.

So whether it's blue or not doesn't seem to be affecting the probability of it being a swirled marble.

And this is because the event of it being a swirled marble is independent of the event of it being a blue marble.

If the probability of it being a swirled marble, given that it's a blue marble is equal to the probability of it being a swirled marble, given that it's not a blue marble, then the event S and event B are independent events.

That means they don't affect each other's probabilities.

Aisha says this means it does not matter what colour marble I select.

There's always the same probability of it having a swirl.

So let's check what we've learned.

A marble is chosen random from the box.

Using the Venn diagram, could you please calculate the two probabilities shown on the screen, and then write down what you notice about your answers? Pause the video while you do that and press play when you're ready to see what the answers are.

The probability that it's a blue marble, given that it's swirled, is 2/10.

The probability it's a blue marble, given that it's not swirled is 10/50.

Now both of those fractions simplify to one fifth.

So what do you notice about your answers? It means the probability of it being blue, given it's a swirl marble, is equal to probability of it being blue given that it's not a swirl marble.

It's also equivalent to the probability of it being blue altogether.

There are 12 blue marbles in a box out of 60 marbles.

All three of those fractions can be simplified to the same thing, one fifth.

That confirms that events S and B are independent.

So it's up to you now for task B.

This task contains three questions, and here is question one.

Pause the video while you do this and press play for question two.

Here is question two.

Pause while you do this and press play for question three.

Here is question three.

Pause while you do this and press play for some answers.

Okay, let's go through some answers.

Here are the answers to parts A and B of question one.

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

And here are the answers to part C and D of question one.

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

Then question two.

Here are the answers.

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

Here are the answers to question 3A.

Pause the video while you check these and press play for the rest.

And here are the answers to question 3B.

Pause while you check these and press play for a summary of today's lesson.

Great work today.

Now let's summarise what we've learned.

A Venn diagram can be used to calculate conditional probabilities.

This may involve sets of events.

The probability of event A, given the probability of event B, is written using the notation P, and the bracket A, a vertical line, and B, where the vertical line means given that.

Venn diagrams allow us to identify which events are mutually exclusive if the probability of A given B is equal to zero.

Or B given A equals zero.

Then events A and B are mutually exclusive.

Venn diagrams can be used to identify which events are independent.

If the probability of A given B is equal to the probability of A given not B, then events A and B are independent.

Well done today.

Have a great day.