Lesson video

Hello, and welcome to today's lesson on using Venn diagrams for conditional probability with me, Miss Oreyomi.

For today's lesson, you'll be needing a paper and a pen or something you can write on and with.

It would help if you put your phone on silence so that you're not easily distracted.

Also, if you can get into a space with less noise and distraction, that would really help you to concentrate for this lesson.

If at any point during the lesson, you wish to pause the video, just so that you could understand what is being said at your own pace, then feel free to do so.

You're also allowed to rewind the lesson, if you want to go back to hear, to repeat what I've said.

When I tell you to pause the video and attempt a task, that is for your own good, and it will be great if you do do that, as you can check if you've understood what I've just taught.

Now, pause the video if you need to go get your equipment, or if you need to get into a space with less noise and distraction, and then resume your video when you are ready to proceed with the lesson.

Okay, I'll try this task involves shaded region.

So can you shade the region in the Venn diagram that represents each of this? So pause your video, shade in the region that represents each of the six questions on your screen, and then press play to resume the video when you're done.

Okay, the first one, if I want to shade in the region that represents B compliment, well, that means everything that is not blonde, that doesn't fit into the set of people with blonde hair.

So it is going to be this.

This is the region of people who don't have blonde hair.

Secondly, if I want to shade in the region of G compliment, so that is people who don't wear glasses, well, I am going to shade in all of this and I'm going to leave out the glasses, the sets that represent the glass, people who wear glasses.

So this is going to be G compliment.

Okay, if I want to shade in the region of the intersection of B and G, that means people with blonde hair and people who wear glasses, it is going to be this middle bit here.

So this is the intersection of people with blonde hair and people with glasses.

If I want to shade in the region that represents people with blonde hair or glasses, well, that's going to be everything in both sets.

So either anyone in this region, could have blonde hair or wear glasses.

So this is B union G.

If I want to shade in the region B compliment as well as the region of G compliment, so I can start by shading in B compliments.

So that's people without blonde hair, it is going to be all of this.

And then not only that, I want people who don't wear glasses, the region of people who don't wear glasses.

Well, it is going to be, this as well.

So the shaded region represents B compliment as well as G compliments.

So B compliment union G compliment.

So the only bit that isn't shaded is here.

This bit here is not shaded.

The rest of the Venn diagram is shaded to represent B compliment union G compliment.

Now, if I want to shade in B compliment and G compliment, that means I am looking for the section that they share, the section that B compliment and G compliment shares.

So if I come over here and I first draw, or I first shade in rather, B compliment, well, I know that it is everything here.

If I shade in G compliment, so that's B, that's G, if I shaded G compliment, it is going to be, everything here.

So which section, what section do they both share? It is the outside region, isn't it? So if I have shaded in, B compliment and G compliment, it is going to be this region here.

That was quite fun, wasn't it? Okay, the labelling of region, and I'll try this task, really helps us to know where we're going to pick our probability of an event happening.

So today's lesson is based on conditional probability.

That is what is the probability of an event happening based on the fact that another event has already happened.

So if I read this out, what is the probability that A happened given that B is already happened? That B is also there.

So I could say, say A is Architecture, for example, and B is Biology.

What is the probability that a student studies Architecture given, this dash here means given, that they also study Biology.

Well, how many students study Biology? It is 10, isn't it? 10 students study Biology.

Out of these 10 students, what is the probability that out of this 10 student, someone also studies Biology, someone also studies Architecture, rather.

Out of this 10 Biology students, what is the probability that someone studies Architecture? Well, it is four, because it is based on this, given out of this 10 students, four students also study Architecture.

So I am going to write four here, right.

Let's go to the second one.

What is the probability that a student studies Biology, given that they also study Architecture? So we're looking here because our condition is based on the fact that they also study Architecture.

So the total number here is going to be my denominator.

What is it? How many students study Architecture? It is seven, isn't it? So out of the seven students, what is the probability of someone who studies Biology out of the seven students? Again, it's going to be four 'cause four is also for Architecture and Biology.

So out of this seven students, four students also study Biology, given that they study Architecture.

Let's move on to this example.

I can see that my circles are separated from each other.

That means the event of one happening does not affect the probability of another event happening.

So we call this mutually exclusive event.

So looking at A, how many students in A? Well, it's one, no, sorry.

Looking at what is the probability of students study Architecture, given that they study Biology? So my denominator is going to be the total number of students in Biology, which is four.

Now out of these four students, what is the probability that someone also studies Architecture as well as Biology? Well, it's none, there's no overlap here.

So it's zero.

So it's zero over four, which is zero.

Now, given that they study Architecture, so my denominator is going to be, one.

Given that they study Architecture, what is the probability that they also study Biology? Again, there's none because there are no overlaps in my circle.

So it's going to be zero, which is zero.

So this is conditional probability, the probability of something happening based on another event.

Okay, let's look at this example, we've got B as a subset of A, remember from previous lesson, we said, B is a subset of A.

If we want to work out the probability of a student that've studied Architecture, student that studies Architecture, given that they study Biology.

Well, the total number of students who study Biology is seven.

And the number of students who study Architecture, given that they study Biology is, is seven, isn't it? Because seven is in A, so it's going to be seven over seven, which is one, right.

Now, the probability of students who study Biology, given that they study Architecture, what's going to be my denominator? Well, we know that this seven is in set A as well, so it's going to be seven plus one, which is eight.

My denominator is eight.

The total number of students studying Architecture is eight.

Now the number of students studying Biology, given that they study Architecture is seven.

So it's seven over eight.

Let's look at this example, that I've got A for students who study Art, B the set for students who study Biology and C the set for students who study Chemistry.

Now, remembering that this one represents the number of students who only study Art.

This two represents the number of students who study both Art and Biology.

This three represents number of students who study Biology and Chemistry and so on and so forth.

So, I want the probability of picking a student who studies Biology, given that they study Chemistry.

So, given that they study Chemistry remains, my denominator would be the total number of students that study Chemistry.

So, I am going to add all the numbers in this region here.

So seven plus five is 12, plus three is 15, plus nine is 24.

Now out of this set, out of this students that study Chemistry, which region shows me the students that study Biology, given that they study Chemistry? Well it will be this three and this five, isn't it? Because yes, five students study all three subjects, but they still study Biology and Chemistry, out of those three subjects.

And the three here shows the students that study both Biology and Chemistry.

So I'm going to be adding these two regions together.

So that's going to be eight over 24.

So the probability of students who study Biology, given that they also study Chemistry.

Now, I want the probability of students who study all three subjects, given that they also study at least two subjects.

Remember here, or rather, I didn't say they only study two subjects.

I'm saying they study at least two subjects.

If I put the word only here, that means they must only study two subjects.

Now, what number? How many students study at least two subjects? Well, two here, they study Art and Biology, five they study Art, Biology and Chemistry, but that is at least two subjects.

Seven is Art and Chemistry, and three is Chemistry and Biology.

So I'm going to be adding these numbers here.

So this should give me 17.

So my denominator is 17 because that's the number of students who study at least two subjects.

Now, out of the 17, what is the probability of students who study all three subjects? So what's the probability of me picking students who study all three subjects? Well, students who study all three subjects is this five over here because it overlaps all three circles.

So it's going to be five over 17.

Now I want to find the probability of students who study Art and Biology, given that they do not study Chemistry.

So given that they do not study Chemistry, well, that means I can't count any of this values.

So I'm looking at other values that students that do not study Chemistry are here.

One, two, three, and six.

So I'm going to add them together.

That gives me nine.

That gives me 11.

And that gives me 12.

Now out of these students that don't study Chemistry, how many students study Art and Biology, given that they don't study Chemistry? It's going to be these two here, isn't it? 'Cause two study Art and Biology, but they don't study Chemistry.

So it's going to be two over 12.

Now I know that I can simplify this to a third and this to one over six, but I'm going to leave it as it is.

Your turn then.

125 students were asked about their pets and the information is shown there.

Find the probability that the student chosen at random has a fish, given that they have a rabbit.

So pause your video now and attempt this task.

Okay, how did you get on with that? Well, I want to find the probability, if I write this again, I want to find the probability of a student chosen at random has fish, given that they have a rabbit.

So the probability of students who's got rabbit, is this region over here, and that is 68.

So my denominator is 68.

Now out of the 68 people that have got a rabbit, how many has got fish? Well it's going to be this, isn't it? This middle section.

So it's going to be 27 over 68.

It is now time for your independent task.

So pause the video and attempt all the questions on your worksheet.

Then once you're ready, let's press play to resume with the lesson.

So pause your video now and attempt all the task on your worksheet.

Right, how did we get on with our independent task? I have put the answers on the screen for you.

So what is the probability of a student taking on dance if they sing, well, dancing is, if they sing, sorry, rather.

So our denominator is 27.

'Cause our condition is, given that they sing, what is the probability that they also dance? And that would be 13 out of 27.

Next, the probability that they study Chemistry, given that they're also taking Algebra.

So Algebra is 150 plus 30, which is 180.

And the probability that they take Chemistry, given that is 30.

And then the probability that they take Algebra given that this study Chemistry is, or our condition is 110, isn't it? 'Cause we're basing it on the fact that they study Chemistry, what's the probability that they also study Algebra? So our Chemistry denominator is 110.

And then the probability of them studying Algebra, given that they study Chemistry is 30.

Over here, this is like the example we did earlier on.

What's the probability that they're studying three subjects, given that they're also studying at least two? At least two means it could be more than two.

So we're adding all the numbers here, here, here, and here.

And that should give us 22.

And the probability that they study three subjects, well, that is that overlap over here, isn't it? 'Cause it's for this set, that set and that set.

So it's seven over 22.

Okay, your explore task.

A and B are two sets.

So set A over here, set B over here.

Each set contains dots, as you can see on your screen.

In the example, the number of dots in A is five, one, two, three, four, five.

And the number of dots in B is six, one, two, three, four, five, six.

The total number of dots in A or B is eight.

So A union B is eight.

Which one of these are possible to draw? So, let's look at dots in A and dots in B.

If I say, I want to draw this and I have three dots in A, so assuming I do this.

And then two dots in, I mean four dots in B, so I've got three dots in A and four dots in B, is it possible for me to have that, so that there's either eight dots in A or B.

Well, it's no, isn't it? 'Cause I've got one, two, three, four, five dots.

So this here would be impossible.

So your task is to explore with this.

Draw the ones that you think are possible, how many different ways are they possible? And if they're impossible, well, you just write impossible here.

So have a go at this now, pause the video and attempt to explore task.

We have now reached the end of today's lesson, a very big world onto you for staying all the way through and completing your independent tasks as well.

So hopefully now you know how to use Venn diagrams to complete or to work out conditional probability.

Do complete the quiz before you go today.

And I will see you at the next lesson, bye.