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 the aim of today's lesson is to be able to calculate theoretical probabilities for combined events using Venn diagrams. So we'll be focusing on two events, which we'll combine together and work out their theoretical probabilities.

On the screen, there are two keywords that you've met in your previous learning.

So one is about theoretical probability and the other is about a Venn diagram.

So it may be that you want to pause the video now just to refamiliarize yourself with those words, and then you can press play when you're ready to continue.

So our lesson has taken two parts, two learning cycles, and the first learning cycle is about theoretical probabilities of combined events, obviously from Venn diagrams, and then we're moving on to further combinations of events in the second learning cycle.

But we're gonna start now with this first learning cycle about combined events and the theoretical probabilities.

Here on the screen, we've got a Venn diagram sorted into two events, and if all the outcomes are equally likely to be selected, then we can work out the theoretical probability of one event.

And in this case, it's the probability of an outcome from event A being chosen.

So we've got our notation with the capital P, which standing for probability of, and then the event is written within the brackets.

We're using A, so this means probability of the event A happening, and this probability would be 5 tenths or a half.

Why is it 5 tenths? Well, there are 5 desired outcomes, the outcomes that are in event A, and you can see them in the region on the Venn diagram, out of 10 in the sample space.

So a Venn diagram always has that rectangle and that rectangle encloses all of the possible outcomes and this is the sample space.

Again, if we wanted to work out the probability and they were all equally likely to be chosen, when we select one from random, what is the probability that we would get an outcome that is in event A and event B? Hopefully you've got an idea what the correct answer is for this, it's 3 tenths.

Why is it 3-tenths? Well, that's because there are 3 outcomes that exist within event A and event B, and we can see that in the intersection of the Venn diagram and it's outta 10 because there are 10 outcomes in the sample space.

Another question from this Venn diagram is what is the probability that we would randomly select an outcome that is not in event A and not in event B? So once again, you need to identify the region of the Venn diagram that shows the outcomes that satisfy, not A and not B.

And this is 3 outta 10.

So where are those 3? Well, we, if we sort of blank out, if we sort of ignore event A and event B, because we would like the outcomes that are not in event A and not in event B.

So they can't be in only A, they can't be in only B, they have to not be in either of those circles.

So if we sort of remove that part of the Venn diagram, then it's that sort of external part that's left, that external region of which there are 3 outcomes, and there are 3 out of 10 because there are 10 possible outcomes in the full sample space.

So this is your check.

What's the probability of getting an outcome that's in A and B, what's the missing numerator? Press pause whilst you decide on what numerator that should be, and then when you press play, you can check your answer.

2 outta 10, there are two in the intersection, you are looking for the outcomes that are both in A and B.

Another check, this time you are answering the question of probability that it's not A and not B.

Pause the video and then press play when you're ready to check.

3, there are 3 outcomes that are outside of both circles.

It needs to be outside of both circles because we are using the word, And.

So Venn diagrams can show two events like we just saw with two circles, they're normally circles, but they can also show 3 and we can still do theoretical probabilities from a Venn diagram that represents 3 different events.

So if we are looking for the probability of A and B, then it is 3 because there are 3 outcomes that are in both A and B that exist within both circles outta 10 because our sample space has got 10 possible outcomes.

But why is the outcome 5 not included? So look where the 5 is, why is the outcome 5 not included? The outcome of 5 does not exist in the event A circle.

And if we are looking for the probability that it is in A and B, the outcomes need to exist in both of those regions.

And that's what Jun's just clarified for us.

So when two events are combined using, and, the outcomes need to satisfy both events.

So let's just check with that, we're now looking at the probability of A and C, whereabouts on the Venn diagram is the region where outcomes exist if they are in A and C? But it's this region and in this Venn diagram there's only one outcome there, and that's number 4, outta 10, because there are 10 in the sample space, so there's one desired outcome out of the 10 possible outcomes.

Does it matter that the 4, that outcome of 4 is an of event B as well? So just think about that.

No, 4 is an outcome of both A and C, and that's the focus of the question, it doesn't matter that 4 is also in event B.

If 4 was not in event B but was still in that intersection, we'd still say the probability is 1 outta 10 because the focus is about it being A and C.

Okay, what's the probability of B and C? So we know it's outta 10 because the sample space is outta 10, but what's that numerator going to be? This one's gonna be 2.

There are two outcomes, there is the outcome of 4, and there is the outcome of 1 that are within the region where the outcomes that are in B and C exist.

You could simplify 2 outta 10 to one fifth or 20% or 0.

2, but we're gonna keep it as this fraction.

What's the probability that we randomly select an outcome that is not A and not B? So whereabouts is that on the Venn diagram? So just by shading out to sort of ignore event A and event B, 'cause we are looking for not A and not B, we can see that there are 3 outcomes that are left in the Venn diagram.

There's the outcome that's in the only C region, and then there are two outcomes that are in the region, which represents all outcomes that are not part of any of the events.

5 and 1 are outcomes that are not in event A, so why have they not been included? So 5 and 1 are outcomes that are not in event A, and we were looking for the probability that we would randomly select an outcome that is not in A and not in B.

And that's because they're in event B.

So remember what Jun said that if we've got a combined event using the word, and, it needs to satisfy both, and 5 and 1 wouldn't satisfy not B because they are quite clearly outcomes of event B.

What's the probability that we would randomly select an outcome that is not A and B? So once again, we know the denominator's gonna be 10 because our sample space has got 10 possible outcomes, and this would be 2 outta 10.

So we need to satisfy both parts.

It needs to not exist in the circle of A, so I've shaded out the A, but it does need, the outcomes need to be in the event B circle.

So there are two outcomes that would satisfy that, that's the outcome of 5 and the outcome of 1.

Why are the outcomes 3, 7 and 9 not included? So just think about that question.

Why are the outcomes 3, 7 and 9, look at the Venn diagram, look at the regions they're in, why are they not included in the probability of not A and B? Because they are not in event B.

This question needs to be outcomes that are exclusively not in A and in event B, 3, 7 and 9 are outcomes that are not in event B.

Another way of looking at this is that we could express the outcomes in each event as a list.

We could list the outcomes, you'd have done listening of outcomes before.

So event A, if we just focus our attention to event A, which is the circle, it's any outcomes in there which is 2, 4, 6, 8, and 10.

We could do the same for B and we can list the outcomes that are within event B, and we can do the same for C.

Say, what events are not in A? And you can do this as the complement to the the list you had previously outta the sample space.

So event A seems to be even numbers 2, 4, 6, 8, 10, and so the not a list is odd numbers 1, 3, 5, 7, and 9.

We can do the same for not Bs, any outcomes and you can see that on the Venn diagram, if you sort of ignore circle B, any outcomes that are outside of event B.

And we can do the same for event C.

So we can list the outcomes If we wanted to to find and identify the number of outcomes that are in event A and not C, then we can look at the two lists and it's those outcomes that are in both of the lists and these are easily identified on the Venn diagram.

So in the two lists, the ones that are in both 2, 6, 8, and 10.

And if you look on the Venn diagram, where do they exist? Well, they exist within the circle of A because it was event A, but not in this circle of C because we were looking for, and not C.

So that outcome of 4 that is in A and C is the one we don't want because it's in C, and we do not want an outcome to be within C for this statement.

So here's a check, which Venn diagram shows the combined event of A and not B? So pause the video, go through those Venn diagrams very carefully, think about what it's saying, and then when we want to check press play.

So it is part B or Venn diagram B that's the bit that you would want.

We were looking for outcomes that are in A but not B.

So the grade out B is is showing you that you don't want it in the B circle, but you do want 'em to be in the A circle.

The first Venn diagram is the sort of opposite of that, which is not A and B, and the last Venn diagram is all the outcomes that are in A.

Here's another check for you, which is for you to finish the fraction for the probability that the outcome would be A and not B.

So pause the video and then when you're ready to check your answer, press play.

There are two outcomes that are in A and not in B.

Remember, it's that region that you just identified on the previous check.

We can find and calculate probabilities if we've got a full Venn diagram, if we've been given the probabilities, then we can work backwards to complete the Venn diagram.

And this Venn diagram is going to represent the number of outcomes as opposed to the individual outcomes.

We've told that the sample space, so all outcomes in this Venn diagram is 65, there are a total of 65 outcomes on the Venn diagram.

Here you've got lots of different probabilities.

So some of these probabilities are for one region of the Venn diagram only, which ones? So just think about that.

So the 3 probabilities of only A, only B and only C, are giving you information about only one region of the Venn diagram, so that seems a sensible place to make a start.

So if we focus on those 3, we can see the numerator is the number of outcomes because the denominator is the total number of outcomes in the sample space.

So they're not in their simplified form, if they do simplify, they're given outta 65 and 65 is the total number of outcomes within the Venn diagram.

And so the numerator is the amount of outcomes for the desired statement.

So for only A, which is that region there we are gonna put 13, and for only B we can put 17, and for only C we can put 12 because it isn't simplified.

So we've used that information, that gives us a helpful start to sort of reverse engineer in the Venn diagram.

Event, A, B, and C, they each cover 4 regions in the Venn diagram.

So if you look at each circle, there are 4 regions within the circle.

So that's not a helpful place to go next, the two combined probabilities might be more useful.

So A and not B.

Well, A and not B covers two regions on the Venn diagram of which we already know one of them, so that seems a great place to go next.

So the probability of selecting an outcome that is in A and also not in B.

So here I've greyed out, I've sort of shaded out the event B because we don't want to select an outcome from B, but we do want to select an outcome that is in A, and so they are the two regions.

We know that one of the regions has 13 outcomes and we got that from the probability of only A, and then there's that second region that we don't know.

So we've called it X.

This means that 13 add X has to equal 18.

Once again, the denominator of 65.

The 65 is the total number of outcomes in the sample space.

So the numerator is the amount of outcomes, there's no scaling necessary.

So we've got ourself a nice, simple linear equation.

We can solve that to say that X therefore has to be 5 in order for the total to be 18, one of the numbers being 13, the other one has to be 5.

Then we can use the probability that it's not A and C, and that is 19 outta 65, so we need to have a total outcomes of 19.

So here we've put the region that we don't have an answer for yet as Y, once again we can set up a linear equation, 12 plus Y needs to total to 19, and I'm sure you know that that means that Y would be 7.

So now we've reverse engineered from that probability, which is a combined of not in A, outcomes that are not in the A circle and in the C circle, which is why it doesn't include 17.

17 is not in the C circle, so we can't include it in this combination.

So we are left with 3 probabilities, which is the probability of event A, a probability of event B, and the probability of event C.

We're gonna use the probability of event C because if you look at that circle of C, there's only one region left that's missing a number of outcomes, whereas for A, there are two regions and for B, there are also two regions.

So the probability that we'd randomly select an outcome from event C is 27 outta 65.

By putting a variable of Z there, we again, we can set up a linear equation.

So 5 plus 7 plus 12 plus Z needs to sum to 27.

If we simplify the numbers that we have that they have a sum of 24 and therefore Z is 3.

That region that we just put the number of outcomes in is part of the event A circle and also part of the B circle.

And so that's really useful for us to work out the last remaining region.

So either of the two probabilities, the probability of A or the probability of B could be used and we'll do that simultaneously to show that.

If we use event A, we can say 13 plus 5 plus 3 plus M has to equal 27, whereas if we used the event B, we'd say 3 plus 7 plus 17 plus M has to equal 33.

So those numbers have come from the probabilities that were given and also from the number of outcomes that we'd already previously placed into the Venn diagram.

And working through both of those linear equations, you end up with M equals to 6.

So once again, it wouldn't matter if you used event A only or event B, you'd get to M equals 6.

So the sample space has got 65 outcomes, the final region of the Venn diagram is the region for outcomes that are not in event A or event B or event C, the part that is outside of all 3 circles.

Are there any outcomes in that region? Because Venn diagrams don't always have to have outcomes in that region, but are there for this one? Yes, and the reason there are is because the sample space needs to have 65 outcomes and currently there are only 63 within the 3 circles.

So that means that there are two outcomes that exist outside of those 3 events.

So a check, the sample space has 40 outcomes.

Given that, the probability of only A is 9 outta 40, the probability of only B is 7 outta 40, and the probability of only C is one fifth, add this information to the Venn diagram.

So press pause whilst you're working through that, and then when you're ready to check press play.

So the probability of only A was 9 outta 40.

So because the denominator was 40, which is the total number of outcomes in our sample space, the numerator was the exact number of outcomes that you should just be placing into the Venn diagram.

And the region that's only A is that part of the circle that doesn't overlap with the other two.

So you should have just been able to place 9 in that section.

The same for the probability of only B, the fraction was out of 40, so the numerator could be placed into the only B part of the Venn diagram.

Probability of only C however, was given as a fraction in its simplest form, so one fifth.

So if we then use an equivalent fraction to one fifth where the denominator is 40, then that would be 8 fortieths.

So we'd then be able to say, okay, the number of outcomes that are only C is 8 out of the 40 in the sample space.

Simplifying the fractions in probability sometimes causes more stages of working because it's useful when the denominator is the sample space size.

Continuing with this Venn diagram then, the sample space still has 40 outcomes.

And given that the probability that B and not A is a fifth, add this information to the Venn diagram.

Press pause whilst you're doing that, and when you press play, we're gonna go through the answers.

So the region that that probability is about is now being identified in purple.

So it's outcomes that are in the circle of B, that that exist within the circle of B, but do not exist within the circle of A.

So we already have the only B part which satisfies part of this, but also there's that additional region that's within the circle of C, but still within the circle of B and not within the circle of A.

And one fifth is equivalent to 8 fortieths, and therefore, that needed to total 8.

If you've already got 7 in the only B region, then that means one more outcome needs to be added.

So here is our first task of the lesson.

First question, you need to use the Venn diagram to find the probabilities.

So press pause whilst you're working through those 4 parts, and then when you're ready for the next question, just press play.

Here's question 2.

Once again, you're working out probabilities from a Venn diagram, but this Venn diagram has got 3 events being represented.

Press pause whilst you work through those questions and then when you press play, we'll move on to Question 3.

Question 3 is one where you need to complete the Venn diagram given the number of outcomes in each event.

So it's partly completed and you've got some probabilities that have been told to you.

So using those probabilities, work out the missing number of outcomes in the final 4 regions.

Press pause whilst you're doing that, and when you press play, we're gonna go through the answers.

Question 1, the answers for the 4 parts are on the screen.

I've given you the answers in in all 3 equivalent forms, so fraction, decimal, and percentage.

There was no requirement for you to do that, but just in case you gave it as a decimal, I wanted you to know that you were correct.

Question 2 was very similar to that in terms of working out the probabilities from the Venn diagram.

The Venn diagram was showing you the individual outcomes and not the number of outcomes.

So it was about counting how many you could see.

These were out of 11, there was 11 numbers within the sample space, and therefore, I've kept that as a fraction and not given the decimal or the percentage values because they would be recurring and then would have a rounding error element.

So the fraction is the most exact way of giving the answer.

Question 3, you needed to use the given probabilities and the given number of outcomes already in the Venn diagram to work out the missing 4 numbers.

So you've got 3, 2, 5, and 5 that you should have added to your Venn diagram.

So you were told the probability of not A was 21 outta 40, and that would allow you to work out that there should be a 5 in the region that is not in A, not in B or in C.

Then you were told that the probability of A and not C was 3 outta 10.

3 outta 10 was in its simplest form, it would be more helpful if the denominator was 40, so multiplying to make it an equivalent fraction was 12 fortieths.

That then allows you to work out that there needed to be the 3 to give you a total of 12.

You already had 9 and then you needed that 3.

And then lastly, the probability of B is 13 outta 40.

So you now had 3 of the regions of B, 3, 7, and 1, that total's 11, you needed to total to 13, so that's why there would be a 2 in that middle region.

And then it needs to have 40 outcomes in total, so you could find the sum of all of the regions and that left 5, which is why a 5 went into that region between A and C.

So the last learning cycle of the lesson is about further combinations of events finding the probabilities from Venn diagrams. So we've used a Venn diagrams by now, and this is a Venn diagram that's got two events, A and B.

Which outcomes are in event A? Well, we can identify them to be 1, 2, 4, 5, and 10.

So I've shaded the event A.

Which outcomes are in event B? Again, we're looking at the full circle event B, and that's 2, 4, 6, 8, and 10.

If you need an outcome that is in event A or event B, which would be okay? So just think about that.

If you needed to select an outcome from that Venn diagram that's in event A or in event B, which ones would you be allowed to select? Well, you could select 1, 2, 4, 5, 6, 8, or 10 as they are either in event A, in event B or in both, and it is the combination of both events.

So if an outcome is to be selected at random from this Venn diagram, we can use it to work out our theoretical probabilities.

So the probability of A or B is 7 outta 10, we just saw that with the listing, but if you'd selected 1, 2, 4, 5, 6, 8 or 10, you would have selected an outcome that's in event A or in event B.

what's the probability that it's not in A or not in B? Well, let's think about this before we get to the probability, which outcomes are not A? Well, it's all of those in the shaded region, we're sort of ignoring the circle of A this time, it's all outcomes outside, so that is 3, 6, 7, 8, and 9.

Which outcomes are not B? So same idea, if we sort of ignore the circle B, it's all outcomes outside of B, that's 1, 3, 5, 7, and 9.

So which outcomes are either not A or not B? Well, that's the combination of both of those, 1, 3, 5, 6, 7, 8, 9.

You can see on the Venn diagram from the shading that it's all of the space that's shaded that would satisfy an outcome in not A or not B.

So what's the probability that it's not in A or not in B? It's 7 outta 10.

Well, Lucas isn't sure about that, he said, "1 and 5 are in event A though?" And Laura said, "But they satisfy the event, not B, so they are valid outcomes." So if one event is satisfied, it's included.

Yes, so when we are using, or, as our combined event word, as long as one of the events either side of the, or, is satisfied, then it would be included in your total.

So here's a check with that one.

This Venn diagram is about pupils and whether they study French or Spanish.

So what's the probability that they study French or Spanish? Pause the video whilst you're working out that probability.

When you press play, you can check your answer.

So the probability that they study French or Spanish is 4 out of the 5, 4 fifths.

And that is because Andeep and Jun study French, so they satisfy the study French part, Sofia studies Spanish, so she satisfies the Spanish part, and Aisha studies both of them.

If you asked Aisha the question, "Do you study French or Spanish?" She would say "Yes, I study both." Does she study French? Yes.

Does she study Spanish? Yes.

So if they study both, they do study the individual subjects.

So now we've got a Venn diagram that's representing 3 different events and this is about pupils and what countries they have visited.

So event A is if they've visited Spain, event B is if they've visited France, and event C is if they've visited America.

So how many pupils have visited Spain or America? Well, Izzy says she finds it easier if she just goes through each individual pupil and asks the question, have they visited Spain or America? So this is Izzy sort of talking it through to herself.

Has Lucas visited Spain or America? Well, Lucas is within circle A, and circle A represents visiting Spain, so, yes.

Has Alex visited Spain or America? Well, he is in the circle of A and also the circle of B.

By being in A means he has visited Spain.

So yes, Jacob.

Has Jacob visited Spain or America? Well, Jacob's in the region, which only means he has only visited France.

So, no.

Has Aisha visited Spain or America? Well, Aisha's like Jacob, she's in the only B part of the Venn diagram, which means that she's only visited France out of those 3 countries, so that would be no as well.

Has Izzy visited Spain or America? So Izzy is within the B circle, so that means she's visited France, she's also within the C circle, which means she's visited America.

So, yes.

Has Andeep visited Spain or America? Well, Andeep is outside of all 3 circles, which means that he hasn't visited Spain, France, or America.

So, no.

Has Laura visited Spain or America? Well, Laura is within the circle of C, she's outside of A and outside of B, but she's in the only C section, which means out of those 3 countries, she has only visited America, so the answer is yes, she has visited Spain or America.

Has Jun visited Spain or America? Well, Jun is within the circle of A, so has visited Spain, within the circle of C, so has also visited America, but outside of the circle of B, so hasn't visited France, but has Jun visited Spain or America? Yes, he's actually visited both.

And lastly, Sam, has Sam visited Spain or America? Well, Sam is in all 3 of the circles.

So Sam has visited Spain, France, and America, which means that they have visited Spain or America because they have visited both.

And therefore, Izzy can now see having gone through all of those pupils that six other pupils have visited Spain or America, some of them have been fortunate to have visited both places, but six have visited Spain or America.

So if a random pupil was selected, what's the probability that they would've visited Spain or America? Well, it'll be 6 out of the 9 pupils, which does simplify to two thirds.

So Izzy's suggestion is, if you're struggling to figure out which ones you're including or which ones you're not, go through them individually.

This clearly works if the number of outcomes is not particularly high, if it was quite a large sample space, then that would take quite a long time, so you then need to start thinking about the regions.

But if it's a lower amount of outcomes, then that can be a really helpful way for you to get through the question.

So there's a check.

If a random people was selected, what is the probability that they have visited France or America? So France is event B, and America is event C.

So if we were to randomly select a people from the nine, what's the probability that they would've visited France or America? Pause the video whilst you figure out that probability, and then when you're ready to check, press play.

7 outta 9.

You are including all pupils that have visited France and all pupils that have visited America.

So there are 7 pupils that satisfied visiting France or visiting America.

Some of them have visited both.

You know with the context of which countries these nine pupils have visited, what's the probability that either they have visited Spain or they haven't visited France? So we're now looking for pupils that have visited Spain or pupils that haven't visited France.

So Lucas, Alex, Sam and Jun are all in the event A, which is those pupils that have visited Spain, so we're definitely want those because they satisfy the visited Spain part.

Now we're looking for those that satisfy the, hadn't visited France, visiting France was event B.

And so that is Laura and Andeep haven't visited France, Jun and Lucas haven't visited France, they are on the outside of the circle for event B.

Lucas and Jun were already included because they had visited Spain.

So if we select Lucas, he would've satisfied the fact that he has visited Spain, but he had also satisfied that haven't visited France.

So he is one that's on both, but that's okay as long as he satisfies one of the two parts of the statement.

So that means there are six pupils that if they were randomly selected, would satisfy they have visited Spain or that they haven't visited France, which simplifies to two thirds.

So here's the checks, same context for you.

If a random pupil is selected, what's the probability that they haven't visited America or that they have only visited France? So think about is this idea for you, go through them individually, if that helps.

If they satisfy one part of the, or, statement, then they are included, if they satisfy both, that is also fine.

So pause the video, and then when you're ready to check press play.

5 out of 9.

The pupils that would be included are Lucas, Alex, Jacob, Aisha, and Andeep.

So you are on your task for this part of the lesson.

So Question 1, use the Venn diagram to answer the following questions.

So you've got some listing to do firstly, and then finding a probability.

Pause the video whilst you do that, and then when you press play, you've got Question 2.

Question 2, you need to use the Venn diagram to answer the following questions, and this one is all about finding the probabilities.

It might be that you want to do it similar to question 1, so listing the outcomes first and then looking to see which ones would be included, a bit like Izzy suggested sort of going through outcomes one by one, it might be that you are getting very comfortable now with which regions you would include.

So pause the video whilst you're working out those probabilities, and then when you press play, you've got one more question on Question 3.

So Question 3, we've got pupils in the Venn diagram, there's 3 events, we don't don't know what those events are, we don't have a context to this question, you need to work out the probabilities.

Press pause whilst you're working out those probabilities, and then when you press play, we're gonna go through our answers to questions 1, 2, and 3.

Question 1, it starts by listing.

So list the outcomes that are in event A.

So you can see within that circle there's outcome 12, outcome 15, outcome 18, and outcome 20.

Then do the same for B, but that was for not event B.

So outside of the circle, which was 16, 17, 18, 19, and 20.

And hence, so hence normally tells you that what you've previously done is gonna support this question, list the outcomes that are in event A or not in event B, so it is combining the two lists together.

If there were any that were in both lists, then they're still included.

And then part D, find the probability that you would randomly select an event that is in A or not B.

Well, your list in part C had 7 outcomes and there were 10 in the sample space.

So 7 out 10, which is 70%, which is nor 0.

7, you only needed to give one of those forms. Question 2, use the Venn diagram to answer the following questions.

So working through those, you got 8 out 10, 7 out 10, 8 out 10 and 6 out of 10.

Once again, I've simplified fractions if possible and given the percentage and decimals.

You didn't need to do that, you didn't need to simplify your fractions in probability, you didn't need to give it as the other alternative forms. But if you gave it as a percentage, you can see that you are correct hopefully.

And Question 3, this one, we don't know the context, but we've got our pupils here.

There are 10 pupils in total, so A or B, so anyone that was in the event A circle, and anyone that was in the event B circle, if they were in both, that's absolutely fine.

So there were 7 pupils for that one outta 10.

Part B was 8 outta 10, C was 8 outta 10, and D was 5 outta 10.

Once again, there's the simplified version of any of those fractions that could be simplified, the percentage and the decimal.

So to summarise today's lesson, you can work out theoretical probabilities with the use of a Venn diagram for two events.

Some Venn diagrams will be representing 3 events.

So if you are asked about two events, you're just focusing on a particular region of the Venn diagram.

When we're combining events, then we might use the word, and, so for example, A and B, and the outcomes that satisfy A and B are outcomes which are in both of those events.

Whereas you might be asked questions which are A or B, and if that's the case, then you are looking for outcomes that are in either event A or event B, it could be that they're also in both.

Really well done today, I hope you've enjoyed that, and I look forward to working with you again in the future.