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Hello, my name is Mr. Southall.

And in today's math lesson we're going to be learning about Venn diagrams, and probability.

But before we begin please make sure that you are in a safe and quiet environment.

If you need to go to a quiet room where you are free from distractions.

If you have a mobile phone please put it on silent so that you're ready to learn.

You'll need a pencil and paper for this lesson, so please make sure that you've got access to those.

And remember that you can pause, and rewind this video at any point, and you will be instructed to pause at any point where we've got an activity for you.

So pause now if you need to to collect any items that you need, otherwise let's begin.

So we're going to have a look at the Venn diagram now, and think about how we can complete the Venn diagram to showcase different information.

Here we have two sets, one for silver coins and one for gold coins, and anything that sits outside of those we can place within the rectangle but not in the circles.

So what I'd like you to do is try and place all the different coins in England using their values inside this diagram.

So we'll have a 1P, a 2P, a 5P, a 10P, a 20P, a 50P, a one pound and two pound coin.

So have a think, and where would you place each of those coins in this diagram? Press pause, complete the task, and then when you're ready to resume press play again.

All right.

Let's have a think.

So, here's what I've completed.

And you can see that the silver coins are your 5P, 10P, 20P, and 50P, and our one and two pound coins are both silver and gold.

So they sit in the middle of this diagram in the intersection of silver and gold.

We also have our 1P, and 2P, they are neither silver nor gold, we call those copper coins, so they will sit just outside of those two sets.

Hope you got it right.

Let's continue.

Here's another task, fill in the Venn diagram with the shapes provided.

So you can see here we have a triangle, an oval, a rectangle, a regular hexagon, sorry, a regular pentagon, a square and a trapezium.

So it says fill in the Venn diagram with the shapes such that set A includes all the four-sided shapes, set B includes all the regular shapes, and therefore the intersection of A and B will be four regular shapes.

And then find the probability of picking a regular shape at random, and the probability of picking a triangle at random.

And if you'd like to do this yourself you can press pause.

I'm going to press ahead and demonstrate how to do this.

So, you can see now that we've placed all the shapes inside the Venn diagram, and in set A, we should only have four sided shapes, and you can see that we have three in there.

We have the rectangle, four sides, the trapezium with four sides, and the square which is in the intersection between A and B, but it's definitely in A, it just means that it's also in B.

And it's in B as well because B is regular shapes, and a square is also a regular shape.

The rest of B contains a regular pentagon, but because it doesn't have four sides it's not going in that intersection, it just sits on its own away from them.

And then outside of those two sets we have any of our remaining shapes that do not have four sides and are not regular, and that's our triangle and our oval.

So that's the first part of this problem completed.

And it says, find the probability of picking a regular shape at random.

So if I was to just pick any shape at random here, the two that are regular are here, okay.

So that means that there are only two possibilities out of how many outcomes, one, two, three, four, five, six possible shapes.

Now we can simplify that to one out of three or one third.

So the probability of picking a regular shape at random is one third.

Find the probability of picking a triangle at random.

Well, we've got a triangle here, there's no other triangles in this group.

So we can just say that there's only one possibility out of six possible outcomes, can't simplify that fraction, so the probability of picking a triangle at random is just one-sixth.

Let's try another example.

The Venn diagram shows information about the students in year 12.

Maths students and law students.

I'm going to use the codes M and L for our probabilities.

Now this is already completed this Venn diagram, so we don't need to construct the diagram we're just interpreting it.

The question says, if a student is chosen at random, workout P.

Now that means the probability that that student has selected maths.

So what is the probability of a student picking maths if I pick them at random from this Venn diagram? Well, again, what we're interested in here is this group, that's all the maths students.

So it's a combination of the ones in the intersection, and the ones outside of the intersection but who are in maths.

So we have 28 and 6.

So 28 plus 6 is 34.

So it's going to be 34 students out of our total, well, our total is that 34 plus all the remaining numbers.

So we've got another 30 there, that's going to be 64 plus the 16 that's going to take us up to 80.

So we have 34 out of 80 students, okay? Now you should be able to identify that we can simplify that fraction to something a little bit neater which we'll do in a moment.

Now, let's have a look at these together.

Now, probability of M and L, the intersection of M and L, that's going to be this part here.

So we've just got 6 students out of 80 which is this part, and that simplifies to 3 out of 40, three-fortieths is our probability.

This third one says the probability of M and not L.

So this little dash here means not.

So this means that we want the probability of someone who is maths but not law, okay? So that means it's going to be this group here.

There's 28 students who took maths who did not also take law.

So that's going to be a simplified fraction of 7 over 20, seven-twentieths.

Finally, we have the probability of the union of maths and law, and that means everything that's in maths and law.

So for this group we're looking at all of these here, okay? So we have 28 plus 6 plus 30 which is 64.

And the 64 out of the 80 because we can't forget about these ones, and that's going to give us a simplified fraction of four-fifths.

Now, it's your turn to have a go.

On this Venn diagram I want you to show whole numbers from 1 to 12.

Now what that means is you're going to list every single number, one, two, three, all the way to 12 inside this diagram.

So try not to get confused here because the numbers represent objects.

And we're going to do two sets within this diagram, the set of even numbers and the set of factors, okay.

And you're going to place those numbers 1 to 12 in that diagram, anything that doesn't sit in F you can put outside of them, but inside the rectangle.

And then I want you to work out those probabilities on the right-hand side.

Pause the video, have a go at the activity, and then press play again when you are ready to continue.

So here we have a solution, and you can see the Venn diagram is now complete, and we should have all the numbers from 1 through to 12 represented.

And the even numbers are on this side, and the factors are on this side, and we've got one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, so everything's in there, we've checked that now.

And everything in the left-hand circle should be even, and everything in the right-hand circle should be the factors, so that's 1, 2, 3, 4, 6, and 12.

And anything in the intersection should appear in both sets.

So we have the two, and the four, and the 6, and the 12, okay? Then we can work out our probabilities.

Probability of something appearing in E, it means it has to appear in this circle.

And there are one, two, three, four, five, six objects in there out of 12 objects, so that's six-twelfths, so which simplifies to one half, we should call them elements.

And the probability of something not appearing in F is these two elements and these four elements, so that six elements there which gives us a half.

And the probability of the intersection of E and F, the intersection of E and F is this part, and you can see that we have four elements in there.

So that's 4 out of 12 which simplifies to a third.

And then finally the probability of the union of E and F means everything in E and F, which is eight elements which simplify to two-thirds.

Let's try another example.

A number is chosen at random from those shown on the Venn diagram.

Find the probability of B, the probability of not B, and the probability of A and B, the intersection of A and B.

Okay, so you don't have to complete the Venn diagram in this question, you just need to interpret it.

Press pause, and carry out the activity, and then press play when you are ready to resume.

Okay.

Let's have a look at the solution.

Probability of B just means anything that's inside this circle.

And we have four elements which is going to be 4 out of 10 elements, because we've got 10 in total in the diagram that simplifies to two-fifths.

Second question says the probability of not appearing in B, well that's these four here plus these two.

So that's 6 out of the 10 elements, which supplies to three-fifths.

And then finally the probability of the intersection of A and B.

Well, that's just this one here 'cause it's this area here.

That's just 1 out of 10 elements, so the probability is a tenth.

Next task, draw a Venn diagram to show the universal set of integers from 10 through to 20.

So that's 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20.

And then the set of even numbers, and the set of multiples of five.

So just like before we're going to need a rectangle, and two overlapping circles to represent this diagram.

Once you've created that diagram, work out those probabilities one through five.

Do be careful, and make sure that you notice that there's a not symbol there, and a not symbol there, okay.

Press pause, carry out the task, and press play when you are ready to resume.

Here's one I made earlier.

So on the right-hand side we have the diagram, and you should be able to see in there all the numbers from 10 through to 20 represented, okay.

And that means that there are 11 elements in my set.

So let's work through these probabilities.

We have the probability of M of something being a multiple of five, okay.

Well, that's going to be 10, 20, and 15 'cause they all appear in the multiples of five.

So that's 3 elements out of 11 elements, so we get three-elevenths as our probability.

Second one probability of E and M, or the intersection of E and M.

So that's 10 and 20, and we're looking at this area here.

So that's going to be 2 out of 11, two-elevenths.

Then we have the union of E and M, that's everything in both the circles combined.

And that is one, two, three, four, five, six, seven elements.

We're missing out the ones that are outside of the circles.

So that's going to be seven-elevenths is your answer.

And then this is where we have those tricky not symbols, the little dashes.

So we have the probability of a not E and M, okay.

So the intersection of not E and M.

So that's just going to be this, okay.

It's not in E and it's in M therefore it's one elements out of 11.

Finally we have the probability of E and M, but it's not E and M 'cause we have that not symbol here, okay.

So if it's not in E and M the intersection of E and M that means it's not in this area here.

Everything that's not in that.

Well, that means we've got four here, five, six, seven, eight, nine, we have nine elements, so that's nine-elevenths.

Next task.

Some students were asked if they played badminton or squash.

B for badminton, S for squash.

The Venn diagram shows information about their answers.

A student is chosen at random.

Work out the probability of B, the probability of S, and the probability of B and not S.

Pause the video and complete the task, and then press play when you are ready to resume.

All right.

Let's go through these then.

Probability of B just means that it has to lie in this circle.

So we have 7 plus 5 which is 12, and we have 22 updates in total, 22 elements in total, so it's 12 out of 22 which is six-elevenths.

Part two we have the probability of S, so like the one before we're looking at a circle but it's the other circle this time, and that's going to be 11 elements out of 22, which is one half.

And finally the probability of B and not S means we're just looking at these ones because they're in B but they're not in S, these ones are in B but they are in S so we discount those.

That means we've got 7 elements out of 22, which does not cancel down, so we just leave it as seven-twenty twos.

All right.

Time for our challenge.

Have a go at this explore task.

It's a little bit more difficult than the other ones, it's a little bit more challenging, and I think it will maybe take you a bit longer.

So do pause the video, have a go at it, and then we'll go through the answer briefly when you press play to resume.

Okay, so if we look at the two tasks, complete the Venn diagram is the first one, and we needed to put numbers into this one.

So it says students who did homework, well, that's this group here.

And so we're going to need to split that 20 according to what's in the other set, and the other set says students who passed.

So 16 of these passed so that's going to go here, and that means the four that didn't pass they're going to go here 'cause they're all the students who did their homework.

And then the other ones are students who passed, so we've already got the 16 who did their homework, but there was also this one here who didn't do their homework, so that person goes here.

And what we're left with, well, we've got 21 students there, so there must be another three that sit here which is those three there, okay.

And in the second diagram it says label the Venn diagram.

So this time we just needed to add the text.

And we can see that we've got a set of four, and a set of 17, okay? So if we're looking for that set of four that must be this group here, okay, the students who didn't do their homework, but it's split, isn't it? Split into a three and a one, which is this three and this one.

So the three who didn't pass are on the outside, and the one who did pass is in the intersection, okay.

So if that's in the intersection then this must be this 16 here.

So that means all of these other people who passed.

So that's how we figure out what this label is, and what this label is.

So that brings us to the end of today's lesson.

Really well done on all the learning that you've done today.

It's fantastic.

And just before we finish have a think about what the most important learning was for you in that lesson.

And maybe take some notes on that too.

Anyway thank you so much for participating today.

Enjoy the rest of your day of learning, and I'll see you again soon.