Lesson video

Hello, I'm Mrs. Lashley, and I'm really looking forward to working with you throughout this lesson today.

So I'm hoping you're ready to make a start.

So in today's lesson, we're going to be working out theoretical probabilities from a Venn diagram, focusing purely on one event.

On the screen are keywords that you will have met before, one being about theoretical probability, and the other about Venn diagrams. So you may wish to pause the video here, read the definitions, and then when you're ready to move on with the lesson, press play.

So our lesson's got two learning cycles.

The first one is focusing on theoretical probabilities from Venn diagrams, and the second learning cycle, we're gonna start comparing probabilities.

So let's make a start on that first learning cycle where we focus on finding the probabilities of one event from a Venn diagram.

So on the screen, we can see a Venn diagram which has sorted two events in the sample space.

Remember that the sample space is everything within the rectangle, within the Venn diagram.

Then we've sorted them into two events, event A and event B.

If all of the outcomes are equally likely to be selected, so they've all got the same probability of being selected, what would be the probability that an outcome from event A is chosen? So which part of the Venn diagram represents event A? You're probably familiar with this notation, but the capital P, and then the brackets, what's in the bracket is the event that you are asking about the probability.

So this is the probability of event A happening, and the probability that this happens is 5/10ths.

That simplifies to a half, but we're gonna keep it in that unsimplified form.

And the reason being, is where did the five, the numerator, where did that five come from? Well, that's because there are five outcomes in event A.

And highlighted on the Venn diagram is event A.

So any outcome within the circle that represents event A is part of event A.

So there's five of them.

And why is the denominator 10? Well, the denominator's 10 because there are 10 outcomes in the sample space.

So there was 10 possibilities when you were selecting one of these outcomes at random.

But we were looking for the chance, the probability that it was in event A, so there's only five of those that could possibly be selected.

Here we've got a different Venn diagram, and this one is about pupils, and whether they study French or whether they study Spanish.

So once again, look at the notation.

So this is asking, in shorthand, what is the probability that we would select, at random a pupil that studies French.

So have a look at that Venn diagram.

Which of those pupils do study French? Well, there's three out of five chance, or 3/5ths, and that's because three of them study French out of the five are represented in that sample space.

So again, we're looking at the event F.

It's the same Venn diagram now, but we've changed the question.

So what's the probability that we would randomly select a pupil that does not study Spanish? Take a moment to look at the Venn diagram.

Which pupils do not study Spanish? Well, this has a probability of three out of five as well, because there are three that are on the outside of the event S.

Here on the screen, we've sort of shaded it out.

They're the ones that we don't want, 'cause we want not Spanish.

And it leaves the pupils that study only French, and one pupil that doesn't study either of them.

So three outcomes in the not in event S, and still the sample space is five.

So here's a check for you about finding the probability, the theoretical probability.

On this one, you need to work out what number, what digit should be in the numerator of the fraction.

So pause the video whilst you work that one out.

And when you're ready to check your answer, press play.

So it's four.

You were looking for the probability that the outcome was in event A, and there were four outcomes in the circle that represents event A.

Here's another one.

So again, pause the video.

Once you think you know what the answer is, press play to check it.

This one was five.

There were five outcomes in the circle that represents event B.

And last one then, so another check.

This one is the probability of not A.

You need to complete the full fraction here, both the numerator and the denominator.

Pause the video whilst you're working that one out, and then when you're ready to check, press play.

So there were six outcomes that were not in event A out of the 10 in the sample space.

If you simplified this to three over five, 3/5ths, then that is correct as well.

So Venn diagrams don't only have to have two events.

You will have seen Venn diagrams that also have three events, and they look a little bit like this.

So every event has an overlap, an intersection with the other two events.

So we can still find theoretical probabilities from Venn diagrams which are representing three events.

So if we ask the question of what is the probability that if you randomly select an outcome from this sample space, that it will be an outcome of event A? Well, the answer is five out of 10, 5/10ths, a half.

And that's because there are five outcomes in the circle of A, irrespective of whether they are also in event B or event C.

If they are part of event A, then we are counting them in this fraction.

And there are 10 outcomes in the full sample space.

So five out of 10.

Okay, so probability of B.

Have a moment to think what you think the answer is before I reveal it.

So it's five out of 10 again, because there are five outcomes in event B, and there are 10 in the sample space.

Okay, what about the probability that we randomly select an outcome that is in event C? This one is two out of 10.

There are only two outcomes in event C, and 10 in the sample space.

And then, what about the probability that we randomly select an outcome that is only in event B? Have a moment to think, look at the Venn diagram.

Which part of the Venn diagram represents only B? Well, it's one out of 10.

There is only one outcome that fits only B, and that is the number five.

And there are still 10 in the sample space.

Probabilities are often given as fractions, but they could be given as decimals and percentages, or the simplified version of the fraction.

So all of the probabilities we just worked out previously, they could be written as a decimal or a percentage.

So the probability of A was five out of 10, because there were five outcomes in event A and 10 in the sample space.

But that simplifies to a half, which is a decimal of 0.

5 and a percentage of 50%.

So if you gave your answer as one of those equivalent forms, it's correct.

There's an absolutely nothing wrong with that.

And the same thing is that you need to understand that probabilities can be given in those forms. Probability of B is gonna be identical, because it's also five out of 10.

Probability of C was two out of 10, so it does simplify to 1/5th.

That is 20% and 0.

2.

The last one is in its simplest form already, and the numerator and the denominator are co-prime.

So it's 0.

1 and 10%.

A check here.

Using the Venn diagram, what is the probability that we would randomly select a pupil that is not in event A? Pause the video, and when you're ready to check your answers, press play.

So A, B and D are all correct answers to the probability that a randomly selected pupil is not in event A.

So there were three pupils that were not in event A.

There were five in the sample space.

So the fraction is free out of five.

That is a simplest form, and then that is equivalent to 60% and 0.

6.

So here we have a Venn diagram that's not complete.

It's missing two outcomes.

If you look through the sample space list, you can see it is the integers from one to 12.

If you go through the outcomes that are currently within the Venn diagram, they're not all there.

So which two are missing? So if we run through the outcomes in the Venn diagram, one is there, two isn't, three is there, four is there, five is there, six is there, seven is there, eight is there, nine is there, 10 is there, 11 is there, and 12 isn't.

So the two that are missing are two and 12.

So remember that the Venn diagram represents the full sample space.

So all outcomes need to be present in the Venn diagram for it to be complete.

So without knowing what event A and event B are, without knowing a list of outcomes for each one, it's impossible to be able to put them into the Venn diagram in the correct space.

You might do it by fluke, you might do it by chance, but you don't know exactly where they should be.

If we were to know the probability of selecting an outcome from each event, would that help us to place the missing two outcomes into the Venn diagram? Well, it would.

So given that the probability of A is one quarter.

So if we were randomly going to select an outcome from the sample space, there is a 25% chance that we would've selected an outcome from event A, because 25% is the same as a quarter, it's equivalent to it.

So how many outcomes are there in event A? Well, look, this fraction is in its simplest form, 1/4, but the sample space is out of 12.

So it'd be more useful if we had this fraction written with a denominator of 12, where the 12 represents all possible outcomes that could be selected.

So using our knowledge of equivalent fractions, we can see that 12 is three times the size of four.

So this fraction needs to be multiplied by three over three.

So one times three, to keep it equivalent, gives us 3/12ths.

So this now tells us that there should be three outcomes in event A, so that the chance, the probability that we randomly select one from event A is 25%.

And if you look at the Venn diagram, there are already three outcomes in event A.

So what does that tell us? Well, that tells us that our missing two outcomes from the Venn diagram do not belong in event A.

Now, if we are also told that the probability of selecting an outcome from event B is a half, 50% chance that you would do that, then what does that imply? Well, again, if we make this a fraction out of 12 so that it's out of our possible outcomes, which is 12, our sample space, that one is multiplied by six.

So to keep it equivalent, our numerator needs to be multiplied by six, and 6/12ths is equivalent to a half.

I'm sure many of you could do that already.

So that tells us that there needs to be six outcomes in event B.

How many are there in event B currently? Well, there's only four in event B.

So the missing outcomes of two and 12 need to go in event B, because there needs to be six.

There's currently only four, and we are missing two from the Venn.

Andeep says, "Well, this means that the two and 12 needs to be anywhere within the circle that represents event B." Laura said, "Not anywhere." Because remember, we found out that event A only has three outcomes, of which there are already three outcomes in event A.

So the two and 12 need to be in the only B region.

And that way, event A still only has three outcomes, and event B has the six outcomes that it needs.

So here is a check similar to that for you.

So given that the probability of A is three out of five, the probability of B is two out of five, complete the Venn diagram.

So pause the video whilst you're working through that check, and when you're ready to check where you should've put the missing pupil, then press play.

So Jacob should've gone in the only B region.

Probability of A is three out of five.

There are already three pupils in the event A circle.

So we needed no more pupils within event A.

The probability of B was 2/5ths, two out of five, so we needed two pupils in the event B, which meant that it had to go in the only B region in order not to increase the amount of outcomes within event A.

By going through the sample space list, you could work out the name, the missing pupil that needed to be added to the Venn diagram.

So this Venn diagram is slightly different in the sense that it shows you the number of outcomes within each event.

So it's not an outcome of five.

It's telling you that there are five outcomes that are only A.

So how many outcomes are there within the sample space? So just take a moment to think about that.

If this Venn diagram is showing you the number of outcomes in each region, how would you work out the number of outcomes in the full sample space? So we're gonna find the sum of the four regions, which is 15.

Well done if you worked that one out.

So how many outcomes are there in event A? Again, look at the Venn diagram and think about that question.

How many? So it's not what outcomes, we're not listing, but how many? So this one is eight.

It's the five outcomes that are only in event A, and the three outcomes that are in both event A and event B.

How many outcomes are there only in event B? So looking at the Venn diagram, how many outcomes are only in event B? This is three.

We do not want to include any outcomes that are also in event A.

So it's the only B part, sort of crescent of the B.

How many outcomes are not in event A? So think about, on the Venn diagram, where would you find the outcomes that are not in event A? And that's seven, there are seven outcomes.

It's the three that are only in B, and the four that are not in B, nor are they in A.

So it's everything outside of the event A circle.

So here's a check.

Again, your Venn diagram is representing number of outcomes.

So how many outcomes are there not in event B? Pause the video whilst you decide on how many there are, and then press play to check whether you've got that right.

17.

The 12 that are only in A and not in B, and the five that are not in A nor in B.

So again, it's all of the outcomes are outside of event B.

So here we've got a Venn diagram again that's representing the number of outcomes in each event, and we are told that there are 18 in the sample space.

So it is missing some numbers.

So you can see it's missing numbers in the only A section and the only B section of the Venn diagram.

So if we're told that the probability that we would randomly select an outcome from this sample space that is not in event A is 13 out of 18, what does that actually tell us? Well, this means that the number of outcomes that are in only B would have to be three.

Because where do we find the outcomes that are not in A? They are the outcomes that are outside of the A circle.

So the ones that are not in A nor B, the 10 that was already given to us, and the outcomes that are only in B.

Because the fraction has a denominator of 18 and the 18 is our sample space size, then the numerator is telling us the number of outcomes.

So 10 add something had to be 13, so that's why it's three.

From that, we can then work out that the other missing number from this Venn diagram is two, because it has to total 18.

We were told that the sample space has 18 outcomes, and currently we had 16 outcomes represented.

So the missing two outcomes would be in that last region, which is only A.

So a check for you to fill some missing numbers on the Venn diagram given some probabilities.

So pause the video whilst you work through that check.

When you're ready to check your answers, press play.

So you should've had four in the intersection, the part that is both A and B, one in the only B region, and six outcomes that were not part of event A nor event B.

The way that you've probably worked through this would be from that first probability of not B.

You knew there needed to be 19 outcomes in total.

The denominator was 24, which is the total sample space outcomes.

So we know that the numerator represents the number of outcomes.

We had that there were 13 in only A, which is part of the not B section, and that means there were six left, and those six would have to be outside of the circle B, and that means they are on the outside of event A and event B.

And then the probability that you would randomly select an outcome from A was 17 out of 24.

So that means that there were 17 outcomes within event A.

If you already know there are 13 outcomes that are only in A, then that means there are four that are in A and B.

And then from that, you can work out that the missing number of outcomes in the only B region was one, because it had to total 24.

So you're gonna do some practise to do with finding the theoretical probabilities from Venn diagrams. So question one is on the screen, there's four parts to it, using the Venn diagram to write the theoretical probabilities.

So pause the video whilst you're working through question one, and then when you come back, we'll go to question two.

So question two is again, writing probabilities, theoretical probabilities, but this Venn diagram is representing three events.

Pause the video whilst you're working through those probabilities, and then when you're ready for question three, press play.

Question three.

For part A, you need to complete the Venn diagram using the given probabilities.

And then for part B, you need to write a probability.

So pause the video, and then when you finish with that one, we've got one more question of this task, and then we'll go through the answers.

So here is your final question of this task.

You again need to complete the Venn diagram, given the probabilities.

So press pause whilst you're working through question four, and then when you press play, we'll go through the answers.

Here are the answers for question one.

I've given all of the answers in the different formats.

So a fraction, a simplified form of the fraction, if it could be simplified, percentage, and a decimal.

There was no requirement for you to do all of those.

But just in case you chose to do it in decimals, I wanted you to be able to see that you were correct.

So have a look at those.

You could always add that onto your workbooks to remind yourself of equivalent forms. So take your time to check through that one.

Question two, same thing.

I've given it as a fraction straight from the Venn diagram.

So number of outcomes that we wanted over the total possible in the sample space.

I've simplified the fraction, if possible, and given the percentage and the decimal.

Here's question three.

So part A, you needed to complete the Venn diagrams by using the given probabilities.

So you needed to have the outcome three, which was missing from the Venn diagram, needed to be put in the only A part of the Venn diagram.

And then for part B, you needed to work out the probability that you would select an only A outcome, and that means it would be two out of eight, which does simplify to a quarter, which is 25% or 0.

25.

Once again, you only need to give one of those forms. You do not need to give all of them.

Finally, question four, you needed to complete the Venn diagram for the number of outcomes in each region.

So there should've been three outcomes in the only A region, five outcomes in the A and B, and six outcomes in the region that represents outcomes that are not part of A, nor part of B.

So we're now up to learning cycle two, which is about comparing probabilities.

Still theoretical probabilities from Venn diagrams, but we're also gonna add an element of comparison.

Aisha and Sofia are playing two different games, and the number of outcomes are shown in the two Venn diagrams. So we've got game one's Venn diagram on the left, and game two's Venn diagram on the right.

Aisha will win if she selects an outcome from an event A in either of the games.

Sofia will win if she selects an outcome from event B in either of the games.

So event A outcomes are part of Aisha's winning, and Sofia gets event B.

So firstly, which game should Aisha play? So for her to have the best chance of winning, the highest probability of randomly selecting an outcome from an event, A, which game would you play if you were Aisha? So Jacob suggests that she should play game one.

So there are four outcomes in event A.

We can see the number of outcomes in event A is four for game one, compared to only three in game two.

So he's saying, "You've got more outcomes in game one.

Play game one.

Why would you play game two?" Jun says, "No, I think it needs to be game two.

Because the probability of getting A is 0.

25, which we know is 25% or a quarter, whereas in game one it's only 0.

2.

Who do you agree with? Jacob because there are more outcomes on game one than game two, or Jun because the probability is higher on game two? So based on probability, you would tell her to play game two.

Yes, there are less outcomes in event A, but there are less outcomes over overall, and therefore her chance of selecting one of her ones is higher on game two than it is on game one.

So which game should Sofia play? Remember, Sofia wins if she selects an event B outcome.

So Jacob's recognised his error last time, so he said, "I need to look at the chance of selecting an outcome from each event B." So Jun says, "Game one, Sophia has a 35% chance of winning, and game two, she has a five out of 12, 5/12ths chance of winning." So which is greater, 35% or 5/12ths? So they've worked out the probability from game one and game two of selecting an outcome from event B, but it's hard to know which one is greater.

We're looking for the one with the better chance.

So we want the highest probability, but how do we know which one's which? Well, this is where the different forms, the equivalent different forms is really useful.

So 35% is equal to 7/20ths.

And to compare fractions, you'd have done this before, a common denominator is what we will need.

7/20ths, there are seven outcomes in event B on game one out of the 20 total in the sample space, is equivalent to 21 over 60.

So both the numerator and the denominator have been multiplied by three.

And in game two we had, the probability was five out of 12, because there were five outcomes in event B and 12 in the sample space, and that is also equivalent to 25 out of 60.

So 60 has been chosen as the common denominator there because it is a common multiple of 20 and 12.

So which is greater? Well, 25 out of 60 is greater than 21 out of 60.

So Sofia is more likely to win on game two.

So a check.

Here are two games.

In which game are you more likely to select an outcome from event A? So pause the video whilst you decide on that, and then when you're ready to check, press play.

So, probabilities.

Game one, there are seven outcomes in event A and 20 in the sample space.

So the probability of selecting an outcome from event A is seven out of 20, 7/20ths.

Game two, the probability you get an outcome from event A.

There are six outcomes in event A out of 15 in the sample space, which simplifies to 2/5ths, but we want to be able to compare them.

So we can change the denominator of five to a 20, because 20 is a multiple of five, and that means this is equivalent to 8/20ths, and therefore, game two gives you the most chance, the highest probability of winning if you needed to select from event A.

So here is the task on comparing probabilities.

Question one, you are comparing the probabilities from each Venn diagram, and the Venn diagrams are representing number of outcomes.

So there's three parts.

You need to compare the probabilities from each of the Venns.

Press pause whilst you do that, and then come back for question two when you press play.

Question two, another comparison from the Venn diagrams, but this time they are Venn diagrams that have three events shown.

So be careful, make sure you're selecting the correct region of the Venn diagram.

Remember, if you need to compare the probabilities, then comparing fractions with a common denominator, or you could convert them to decimals, or you could convert them to percentages in order to compare them.

But when they're in different forms, it's much more challenging to compare.

Press pause whilst you work through question two, and then press play and we'll have question three, the last question of this task.

So in this question, you need to complete the second Venn diagram with the number of outcomes, such that you are more likely to select an outcome from event B in Venn two.

So think about this one.

You need to complete the Venn diagram so that the probability of selecting an outcome from event B is higher on Venn two than Venn one.

Press pause whilst you're working through question three, and then when you press play, we're gonna go through the answers to this task.

So question one had parts to it.

We've got part A on the screen right now.

So you needed to work out the probability, from doing it from Venn one, the probability on Venn two, and then compare them.

So the first one was the probability of selecting an outcome from event A.

So the probability for Venn one was 11 out of 30, if you just removed the number of outcomes from the Venn diagram.

On event two, it was eight out of 20.

So as fractions, these are not easy to compare.

So we need a common denominator, and 30 and 20 are both factors of 60.

So 60 was a good choice, but not the only choice you could've made.

So we've got our equivalent fractions there, 22 out of 60 and 24 out 60.

So it shows that event A is more likely in Venn two.

24/60ths is greater than 22/60ths.

So this is part B of question one.

Again, you've got to do the same sort of process.

We were looking for the only A, so the outcomes that were only in event A.

So on Venn one, it was eight out of the 30 in the sample space.

On Venn two, it was five out of the 20.

Once again, you can use the denominator of 60, 16/60ths and 15/60ths.

So you are more likely to select an outcome that is only in event A on Venn one.

Here's the last part of question one.

So this one was the probability that you select an outcome that is not in event B.

So there are 20 outcomes on Venn one that are not in event B.

And on Venn two, there were 11 that were not in event B.

But it was 20 out of the 30, compared to the 11 out of 20.

So once again, we're using our denominator of 60, and we get 40 out of 60 compared to 33 out of 60.

Well, 40 out of 60 is greater than 33 out of 60.

So you are more likely on Venn one.

Question two, the probability of selecting an outcome from event A.

So on Venn one, the probability of selecting outcome from event A was the 10 outcomes that are in event A over, or out of 24, which was the sample space size.

Venn two, it was 11 out of 32.

So once again, we've got different denominators.

If you're gonna use fractions to compare, we need a common denominator, and 96 is the lowest common multiple between those two numbers.

Comparing them, it is more likely that you will select an outcome from event A on Venn one than Venn two.

Part B, the probability of selecting an outcome from C.

So C on Venn one was 40 out of 96, and on Venn two it was 24 out of the 96.

So you are more likely to select an event C from Venn one.

And the last part to this question, if you were looking for the probability of selecting an outcome that is not in event B, in Venn two, you've got a better chance, the higher probability, only by one out of 96.

So not a huge difference between those probabilities, but theoretically, you are more likely to select an outcome that is not in event B from Venn two than from Venn one.

And lastly, question three.

So question three, you needed to place a number into the only B region of Venn two, and that number could be any integer greater than four.

So five is the minimum that you could've put in there.

It had to be an integer, because it was about number of outcomes.

So you'd only ever be able to count them in integers.

So any number greater than four.

So five, all the way up to infinity, would make it that the probability of selecting an outcome from event B is greater on Venn two than on Venn one.

So to summarise today's lesson, where we were looking at theoretical probabilities from Venn diagrams, but only focusing on one event, theoretical probabilities can be found using a Venn diagram.

The important part is to focus on the correct region of the Venn diagram.

It may be possible to calculate where missing outcomes can go into a Venn diagram, if you have enough probabilities known to you.

Really well done today.

I look forward to working with you again in the future.