Lesson video

Oh, hello, and thank you for joining me for a lesson today.

My name is Dr.

Rowlandson 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 Set Notation and by the end of today's lesson, we'll be able to use formal language for set notation and see how it relates to various representations.

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 the words mean and then press play when you're ready to continue.

The lesson is broken into two learning cycles.

In the first learning cycle, we're going to learn about set notation by relating it to Venn diagrams. And in the second learning cycle, we're going to apply what we've learned to other representations.

Let's start off with set notation with Venn diagrams. Notation can be used to define sets or write statements about probability in a way that is shorter than writing full sentences.

Let's take a look now at some of the notation we're going to be exploring in today's lesson.

All these will be explained and exemplified in more detail later in the lesson, but let's just see now what they look like.

One of them is the letter P, followed by some brackets and that means probability.

And the thing you write inside the brackets is the outcome or event that you are writing the probability for.

For example, you might put P and then in brackets heads, which means the probability of getting heads when you flip a coin is such a thing.

Or you might put P and A, which means the probability of outcome A happening.

The next one is this Greek letter which is Xi, which represents the universal set and one way you could write that is Xi equals and in some curly brackets write down what that universal set includes.

For example, the positive integers.

The next one looks a little bit like an apostrophe.

It's a prime symbol, but it doesn't have anything to do with prime numbers.

The prime symbol represents the complement of a set.

So for example, if you've got set A, which may include some outcomes, A prime are all the things that are not in set A.

This symbol, which looks a bit like a lowercase U but without the stem, is the union of two sets.

And you write that as, for example, A union B to mean the union of sets A and B.

And this symbol, which looks a little bit like a lower case N without the stem, is the intersection of two sets.

So you could write A intersection B to mean the intersection of those two sets.

Let's take a look at what all these mean by looking at an example.

Here we have Lucas who rolls a 20-sided dice.

The sample space we can now see on the screen shows all the possible outcomes when he does that.

Here they are, all the integers from one to 20, and we've got that symbol there.

The Xi symbol represents a universal set.

In this case, it's all the possible outcomes when rolling the 20-sided dice.

The universal set can also be represented in a Venn diagram.

We've got our rectangle, which is labelled with Xi, and every number from one to 20 is in that rectangle.

Now Lucas wants to roll an even number, so set A represents all the outcomes that would satisfy this event, and they are the even numbers between one and 20.

The Venn diagram can now be adapted to show which elements of the universal set are also members of A.

And it looks something a bit like this.

We've got a circle or circular shape labelled A.

You can see that all the even numbers between one and 20 are inside that, and all the numbers that are not even are still inside the rectangle 'cause they're still belonging to the universal set, but they are outside the circle.

A prime, or A and that dash looking symbol, is the complement of A.

This set contains all the elements of the universal set that are not in A.

In other words, all the odd numbers that are on that dice.

So it'll be these numbers here, the numbers which are now highlighted in the Venn diagram.

A prime can be referred to as the complement of A, not A or A prime.

Now we've learned some set notation, let's look at how we can use it when working with probabilities.

Set notation can be used with notation for probability when finding probabilities of events.

For example, this means the probability that Lucas rolls a number on the dice that belongs to set A.

In other words, the probability he rolls an even number, but it's written much shorter as P and in brackets A equals and we can see the probability will be 10/20, which is 1/2.

This notation means the probability that Lucas rolls a number on the dice which is not part of set A.

In other words, the probability he does not roll an even number, and that again would be 10/20 or 1/2.

So let's check what we've learned.

Here we've got Sofia who rolls a 20-sided dice and the universal set shows which numbers she can roll.

She wants to roll a multiple of three.

Set B represents all the outcomes that would satisfy this event.

Could you please write down all the members of set B? Pause while you do it and press play for an answer.

They are three, six, nine, 12, 15, and 18.

So now we've defined the universal set and set B.

Could you please write down all the members of B prime, the compliment of B? Pause while you do it and press play when you're ready for an answer.

You should have written all the integers between one and 20 that are not multiples of three, and here they are.

All these numbers are in the universal set, but they are not in set B.

Could you now please draw a Venn diagram which looks a bit like the one on the screen here to show which elements of the universal set are members of B and which are members of B prime? Pause the video while you do that and press play for an answer.

Here's what your answer should look like.

All the numbers which are multiples of three should be inside your circle or circular shape, which is labelled B, and all the others should still be inside the rectangle but not inside the circle for B.

Could you now please use this to find some probabilities? Find the probability that Sofia rolls a number which is in set B and find the probability she rolls a number which is in set B prime.

Pause the video while you do that and press play for an answer.

The first probability which is the probability she rolls a multiple of three, is 6/20.

The second probability, which is a probability that she does not roll a multiple of three, is 14/20.

Let's now introduce some more set notation.

We have our universal set defined, which is the integers from one to 20, and we've also defined set A as all the even numbers from the universal set and set B as being all the multiples of three from the universal set.

This notation A union B is the union of sets A and B.

This set contains all the elements of the universal set that are members of either set A or B or they could be both.

To be part of A union B, you just need to satisfy at least one of those two conditions, either be an even number or be a multiple of three.

And that would include all these numbers you can see here.

Set notation can be used as a shorter way to write probability statements.

For example, take this statement here.

The probability that the dice shows a number that is either an even number or a multiple of three is 13/20.

That statement is perfectly fine, but it's quite long.

It could be written much shorter like this.

This means the exact same thing.

We've got P, which represents probability.

We've got the union of A and B in there, which is what's described in a sentence, and we've got what the probability is, 13/20.

Those two things mean exactly the same, but with notation it's a bit shorter, a bit more concise.

Let's introduce another symbol.

A intersection B is the intersection of sets A and B.

This set contains all the elements of the universal set that are members of both A and B.

They have to be members of both sets to be included in the intersection.

They are the numbers where the two circles overlap in the Venn diagram.

They are six, 12, and 18.

Once again, set notation can be used as a shorter way to write probability statements.

Take this statement for example.

The probability that the dice shows a number that is both an even number and a multiple of three is 3/20.

We could write that in a much shorter way like this.

So let's check what we've learned.

Here's a symbol, what does it mean? Pause while you choose and press play for an answer.

The answer is C, the union.

In which Venn diagram is the union of A and B shaded? Pause while you choose and press play for an answer.

The answer is B.

Which symbol here represents the intersection of two sets? Pause while you choose and press play for an answer.

The answer is C.

So could you please draw a Venn diagram and shade in the region that represents the intersection of A and B? Pause the video while you do it and press play for an answer.

It would be this section here where the two circles or circular shapes overlap.

Let's now define a universal set as being all letters from A to Z in the alphabet, set L as being all letters from the word Lucas, and set S as being all letters from the word Sofia.

Could you please write down the members of the set you can see on the screen here? L intersection S.

Pause while you do it and press play for an answer.

You're looking for the letters which are in Lucas and are also in Sofia.

They have to be in both, and they are letters A and S.

Now we've learned some set notation, let's look at how we can combine notation to describe statements which are a bit more complex than the ones we've previously seen.

Multiple set notation can be combined to represent specific sets.

For example, here we have A intersection B prime.

Now that means the intersection of set A and the complement of B.

In other words, all the elements that belong to set A and also do not belong to set B.

It would be these elements which are now shaded.

These are the elements of the universal set that are both a member of A and are also not a member of B.

It would be these numbers here.

This notation, A union B prime, is the union of set A and the complement of B.

In other words, anything that is either in set A or not in set B belongs to this set.

Anything that satisfies either of those conditions or both belongs to the set.

They are all the elements that are either members of A or not a member of B, and these are the ones which are now highlighted.

It's all these numbers here.

So let's check what we've learned.

What set is shaded in the Venn diagram? Pause while you choose and press play for an answer.

The answer is A prime or the complement of A.

The things that are shaded is everything that is not in set A.

So what set is shaded now in the Venn diagram? Pause while you choose and press play for an answer.

The answer is B, A prime intersection B.

We've shaded in the section of B that is not in A.

In which Venn diagram is A union B prime shaded? Pause while you choose and press play for an answer.

The answer is C.

Everything that's in A is shaded and everything that is not in B is shaded as well.

So let's define the universal set as all the letters from A to Z in the alphabet.

Set L are all the letters from Lucas, S are all the letters from Sofia.

Could you please write down the members of L intersection S prime? Pause while you do it and press play for an answer.

You are looking for the letters which are in Lucas and are also not in Sofia.

They are L, U, and C.

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

This task includes three questions and here is question one.

Pause while you do it and press play for question two.

Here is question two.

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

And here is question three.

Pause while do it and press play to start going through some answers.

Okay, let's go through some answers.

For question one, here they are.

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

Then question two, here are the answers.

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

Then question three.

This is a little bit more complex because we have three sets in a Venn diagram.

Let's go through it together.

For the first one, you wanna shade in the intersection of A, B, and C looking for where all three of those sets overlap in the Venn diagram.

That'll be this region in the middle.

In the middle one, you are looking for the union of A, B, and C.

You want to shade in everything that's in A, everything that's in B, and everything that's in C.

That'll be all of this.

And the third one, looking for the intersection of the complement of A and B and C.

You could think about this as looking for the region where B and C overlap, but it doesn't overlap with A, and it'll be this region here.

Well done so far.

Let's now move on to the next part of this lesson where we're going to apply what we've learned about set notation to other representations.

Here we've got Lucas, which has his 20-sided dice, which he's going to roll.

All the possible outcomes are now represented in a table.

You should be able to spot all the integers from one to 20 in that table somewhere, but they are organised in such a way that the columns show you which ones are even and not even, and the rows show you which ones are multiples of three and which ones are not multiples of three.

By defining sets A and B, set notation can be used to identify elements belonging to specific sections of the table.

For example, we could define set A as being all the outcomes which are even and set B as being all the outcomes which are multiples of three.

And then we can use set notation to describe particular sets and also write probability statements.

For example, we could put A, that is all the numbers which belong to set A.

That'd be these numbers here, the even ones.

And the probability of rolling a number from set A would be 10/20 or 1/2.

Or we could put A prime and that would be all the numbers which are not in set A, all the numbers which are not even, and they'll be these numbers here.

And the probability of rolling a number that belongs to the complement of A would be 10/20.

Or we could put B, which is the set of numbers which are multiples of three.

These numbers here.

And the probability of rolling a number which belongs to the set would be 6/20.

Or we could write more complex ones such as this one, A intersection B.

We're looking for the numbers which belong to A and also belong to B, all the ones which are even and multiples of three.

These numbers here.

And the probability of rolling a number which belongs to the intersection of A and B would be 3/20.

Or we could have the union of A and B.

They are all the numbers that are either in A or in B or in both.

It'd be these ones here.

We've highlighted the entire column for even and we've highlighted the entire row for multiples of three.

The probability of rolling a number from this set would be 13/20.

So let's check what we've learned.

Which table shows B prime shaded, the compliment of B? Pause the video while you choose and press play for an answer.

The answer is C.

That table's highlighted the rows which are not in B.

Which table shows the intersection of A and B prime shaded? Pause while you choose and press play for an answer.

The answer is B.

That table shows just the one cell shaded and that cell is in A and it's also not in B.

Which set describes the shaded region in this table here? Pause while you choose and press play for an answer.

The answer is D.

That table has shaded all the cells that are in the column for A and all the cells that are in the row for B prime.

So it's the union of A and the complement of B.

Here we have Lucas who rolls a 20-sided dice.

Set A are the even numbers.

Set B are the multiples of three, and they are shown in the table you can see in the screen.

Could you please find the probability that Lucas rolls a number that belongs to the set you can see on the screen there? Pause the video while you do it and press play for an answer.

The answer is 17/20.

The set we're looking for is a union of A and B prime or the things that are either written in A or in the complement of B.

In other words, either it needs to be even number or not a multiple of three.

Let's now apply what we've learned about set notation to probability trees.

Lucas rolls a 20-sided dice.

We've defined event A as being he rolls an even number and event B as being he rolls a multiple of three.

And combinations of events are represented in this tree diagram we can see here.

We can see the first time we're trying to decide whether or not it's even or not even.

And then the second layer of branches, we're trying to decide whether or not it's a multiple of three or not a multiple of three.

And the probability of each case is written on the branches.

Set notation can be used to identify specific combinations of events in the tree diagram.

For example, we could look at this combination of events here.

That's Lucas rolls a number that belongs to the intersection of A and B.

In other words, he rolls a number that is even and a multiple of three.

It would be this situation which is now highlighted in the probability tree.

He rolls a number, it's a even number, and he rolls a number that's a multiple of three.

We can find the probability of this happening by using multiplication.

Multiplying 1/2 by 3/10 to get 3/20.

Here's another example.

The probability that Lucas rolls a number that belongs to the intersection of A and B prime.

In other words, he rolls a number that is even and not a multiple of three.

It would be this combination of events which are now highlighted on the screen.

That probability would be 1/2 multiplied by 7/10, which is 7/20.

This next combination of events, it's a little bit more complicated.

It's the union of A and B.

In other words, the probability that Lucas rolls a number that is either even or a multiple of three or both, it would be these combinations of events in a tree diagram.

Either it's even and a multiple of three or it's even and not a multiple of three or it's not even but it is a multiple of three.

Any of those combinations would satisfy this event.

And to find the probability, we need to multiply each pair of events together and then add all those probabilities together afterwards.

And then we'll get 13/20.

So let's check what we've learned.

In which diagram is the intersection of A prime and B highlighted? Pause the video while you do that and press play for an answer.

The answer is B.

Which combination of events is highlighted in the tree diagram here? Pause while you choose and press play for an answer.

The answer is C.

Use the tree diagram you can see here to calculate the probability of A intersection B.

Pause while you do that and press play for an answer.

This is the event that A happens and also B happens as well, which you can calculate by doing 1/3 multiplied by 2/5 which is 2/15.

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

This task contains three questions and here is question one.

Pause while you do it and press play for question two.

And here is question two.

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

And here is question three.

Pause while you do this and press play to go through some answers.

Okay, let's go through some answers.

Here are your answers to question one.

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

And here are your answers to question two.

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

And here's question three.

This one involved a frequency tree.

Now we haven't looked at a frequency in today's lesson, but you can apply what you've learned about set notation to here in a similar way.

Let's go through this one together as it's something a bit different.

A mechanic fixes cars and vans of people who live in both Oakfield and Elmsleigh.

The frequency tree shows the information about the vehicles fixed during a month.

And what you have to do is write down the number of vehicles that belong to each of the following sets.

A, which is defined as vehicles fixed in Oakfield, that would be 85 vehicles.

B, which is defined as cars which are fixed, that would be the sum of 48 and 42, which is 90.

The 48 are the cars that are from Oakfield, and the 42 are the cars that are from Elmsleigh.

But we want all of it together because we just want any cars that are fixed.

That'll be 90.

Then the intersection of A and B will be 48.

We are looking for the vehicles that are from Oakfield and are also cars.

We can see that in the frequency tree.

For D, you are looking for the union of A and B.

So you're looking for any vehicle that is from Oakfield and any car.

Now, you may be tempted to add together your answer from A and your answer from B, but just be careful of that because your answer for part A includes some of the cars which are included in your answer for part B.

In particular, there are 48 cars which you could end up counting twice if you add together your answers for part A and B.

Therefore, you're better adding them up individually, doing 48 plus 37 plus 42, and that'll give you 127.

Or you can take your answer from part A, all the ones that are in Oakfield, and just add the cars that are from Elmsleigh, and that'll be 127 as well.

For E, you want the union of the complement of A and the complement of B.

So in other words, you wanna count anything that is not from Oakfield and you wanna count anything that is not a car.

That would be 42 plus 13 plus 37, which is 92.

And for part F, you want the intersection of A complement and B complement.

In other words, you're looking for all the vehicles that are not from Oakfield and are also not cars.

In particular though, you are looking for vans from Elmsleigh, and that would be 13.

And then for parts G to J, you had to work out some probabilities based on a scenario that a customer's chosen at random for a prize.

For parts G and H, you could use your previous answers to help you.

For example, the probability that the customer is from the intersection of A and B.

That would be 48/140.

You previously worked out how many customers belong to that intersection in a previous question.

In H, you're looking for the probability that the customer is from the union of complement of A and complement of B.

You previously worked out how many customers that was and you have 92/140.

I and J, you haven't previously worked out anything that will help you here.

So you need to first work out how many customers you're talking about and then write it as a probability.

So the probability that the customer is from the intersection of A and B complement, it means you're looking for the customers that are from Oakfield and do not have cars.

So it'll be 37/140.

And for J, the set is the union of the complement of A and B.

In other words, you're looking for customers that either are not from Oakfield or have a car or both.

And that would be 103/140.

Great work today.

Now let's summarise what we've learned.

Set notation is a short way of writing probability statements.

A prime is the compliment of set A.

In other words, all the things that are not in set A.

A union B is a union of sets A and B.

In other words, all the elements that are either belonging to set A or belong to set B or both.

A intersection B is the intersection of sets A and B.

In other words, all the elements that belong to both sets.

And set notation can relate to specific aspects of a Venn diagram or to a table or a probability tree or a frequency tree.

Well done today, have a great day.