Lesson video

Hi, my name's Mrs. Wheelhouse, and welcome to today's lesson on Comparing Representations Of Outcomes For Two Events.

Today's lesson falls in the unit, Probability: Possible Outcomes.

I love probability, so I'm really looking forward to doing today's lesson with you.

Let's get started.

By the end of today's lesson, you'll be able to identify the same outcome across a variety of different representations.

Now, here are some keywords that you may or may not be familiar with.

If you've seen some of the previous lessons in this unit, then these will all be familiar to you.

This is just a quick reminder.

We have outcome table, tree diagrams, Venn diagrams, and outcome trees.

Now, feel free to pause if you want to take some time to review these.

If you're ready though, let's get going.

We have two parts of today's lesson.

So, we're gonna begin with part one, which is transferring information between diagrams. Here we have a menu, and you can see there are two possible starters and three possible main courses.

Now, Jun's going to choose a starter and a main course.

What are the possible outcomes of that selection? So, what pairings could we have? Well, Jun says, "I'm able to take the information about the stages in event, like choosing a starter and a main course and then list all the possible outcomes, or we could place them in an outcome table, or in an outcome tree." So, there's quite a lot of ways that Jun can represent all the different possible choices he has.

But what you want to know is, s it possible to use one diagram to finish a different one that isn't completed, even if I don't have that original information to start with? Hmm, what do you think? Well, Sophia thinks it is possible.

She's saying that she can use the headers of an outcome table to tell us what the menu items are.

Now, luckily, Sophia's gonna give us an example, so let's see what she means.

So, these headers tell you the starters.

Ah, okay, I remember these.

That was the T and the G.

And these headers up here, these tell us our mains, so that's the P, the F, and the S.

"Ah," says Jun, "so I can use these headings on the first layer of the branches." Well, that makes sense, there are only two possible outcomes because there are two branches, so the T and the G could go there." And if you look at the order Jun's put them in, can you see he's put T at the top and G at the bottom? And if we look at the sample space all the way to the right of the tree, that makes sense because we can see outcomes that start with T at the top and ones that start with G at the bottom, so this makes sense they're in that order.

Now, the P, the F, and the S, we know these are the headers of the second layer of branches and there are three, three branches, that makes sense.

Can you spot the order we've got to put them in/ Now, we only have to put them in this order because of our sample space.

So, one of you saw that P is at the top, S is at the bottom, so F is gonna have to go on that middle branch.

And in fact, we either knew that from looking at the sample space or we could have looked at the branches just underneath.

Sofia says, "That's right! Now your outcome tree is full, you can complete the sample space." And you can see that we filled in the missing possible outcomes now.

So, let's do a quick check.

A coil will be flipped and a spinner will be spun.

The outcome table and the outcome tree both show the same trial, so what's gonna go in the two gaps on the outcome tree? Now, you might think there are quite a few gaps, but I've specifically labelled the two gaps I want you to focus on, so that's where you can see the a and the b.

What do you think is gonna have to go there? Pause while you write this down.

Welcome back.

Did you spot what it had to be? It's right, it had to be heads and tails.

Now, it had to be heads and tails because when we look at our headers, what begins our two rows are heads and tails, but we have three columns, so that's three possible outcomes.

So, it had to be heads and tails that went here.

Now, remember, you could have written them the other way round, so tails at the top and heads at the bottom, you'd still be right.

What would go in the two gaps now on the outcome tree? So, you can see I've updated the outcome tree, I've got the heads and the tails there, but what's gonna go in the new a and b positions? Pause the video while you write this down.

Welcome back.

Did you spot that We had to look at the headers for the columns in order to complete this? So, well done if you spotted it, it has to be a 1 and a 7.

Now, at this point remember, you could put these either way round, so it could be 7 at the top for a, and then 1 in the middle for b.

Look at a different outcome table now.

So, spinner one has three outcomes, spinner two has four outcomes, and each spinner will be spun once.

What would go in the three gaps on the outcome table? So, for a, b, and c, what would go in those spaces? Pause the video now while you work this out.

Welcome back.

Did you use the information in the outcome table to complete this? If you did, then you should have worked out that a had to be Y because every outcome in this row has a y in it, and therefore for b, it had to be, that's right, z because every outcome has a z there in that row.

And then c, did you work out what had to go there? That's right, it had be a 3 because every outcome in this column has a 3.

We've got an outcome table and outcome tree now both showing the same trial.

The incomplete sample space has been generated from the outcome tree.

What I'd like to know is what goes in the four gaps.

So, there are three gaps from the tree, and there's one specific gap in the sample space that I'd like you to complete.

Pause while you do this.

Welcome back.

Have you filled these in? Well, by using the sample space, you'll be able to work out what has to go where.

So, a had to be Z because if you follow the branches along and I follow, for example, the branch for Z and then the branch for 0, I can see in the sample space lined up with that top branch is Z0.

So, that tells me Z had to go there, and therefore I know that b has to be X.

To fill in c, I had to look at the headers for my columns in the outcome table and I would've seen, I've already got 0, 1, and 5 in the tree, but I'm missing 3, so need to fill that one in.

And then, we could follow those branches along in order to fill in the sample space.

So, to get to where d is, I have to have gone along Y and then along 3, so it should be Y3.

Well done if you got this right.

Now, it is possible to use one correct diagram to check any errors of a different incorrectly completed one.

In other words, we can use one to check the other.

Alex is going to spin a spinner and it has the following outcomes: 1, 2, 3, 4, 6, 7, 9, and 10.

Alex considers the event that the spinner lands on an even number and he considers the event it lands on a factor of 18.

He shows these events correctly on this Venn diagram.

So, there's our Venn diagram, and you can see that Alex has correctly placed his outcomes.

So, Jun's now going to spin the same spinner as Alex, but he decides to show the events of the trial on an outcome tree instead of a Venn diagram, and you can see what he's got here.

Now, Alex's Venn diagram is fully correct, but Jun's outcome tree is not.

Take a moment to compare the two diagrams and see if you can find the three errors that Jun has made.

Do this now.

Welcome back.

Did you find the three errors? Well, the first one was that 6 is even, and Jun said, "That 6 is odd," but that's not right, is it, Jun? 8 isn't a factor of 18, and I don't know where Jun's got 8 from because that wasn't even a potential outcome from our spinner, you're just making numbers up now.

Now, four is even, but it's not a factor of 18, so Jun's put that one in the wrong place.

So, here we go, we've correctly updated the outcome tree now and everything is where it should be.

Let's do a quick check.

Andeep's going to roll a dice with integers on it from 20 to 29.

Andeep and Laura look at the events that the dice lands on an odd number and the event that it lands on a multiple of 7.

Which outcome on the Venn diagram shouldn't be on the diagram at all? Pause the video while you find that incorrect outcome.

Welcome back.

Did you spot the mistake? That's right, 14.

14 is not in our outcome table and it shouldn't be there on the Venn diagram, so well done if you spotted it.

Now, which outcome on the Venn diagram is in the wrong place? It's a different question this time.

So, there's an outcome that's in the wrong place.

Pause while you find it.

Welcome back.

So, which one's in the wrong place? Well, that would be the 28.

28 is not odd, so there's no way it should be in the intersection of the two circles.

28 should be there because it's a multiple of 7 and it's even, not odd.

Last question now, which outcome on the Venn diagram is missing? So, we've had one outcome that was there that shouldn't be, and we dealt with the 14, and we've had an outcome that was in the wrong place, that was the 28, this time we have an outcome that's missing.

Which one's missing? Pause and make your choice.

Welcome back.

Did you find the missing outcome? That's right, it's 26.

Remember, the numbers that are even and not multiples of 7 have to go outside of our events, but still within our sample space.

We already had 20, 22, and 24, we were just missing the 26.

Now, Lucas is going to roll a dice with integers from 42 to 49 on it.

Lucas and Izzy look at the event the dice lands on an odd number and that it's a multiple of 7.

Lucas correctly shows the outcomes of these events on an outcome tree, and he uses it to help Izzy fill in her Venn diagram.

Which outcomes would go into the region marked a? Pause the video while you write this down.

Well, for a, we need values that are not multiples of 7, but they are odd, so it should be 43, 45, and 47.

Which outcomes go into the region marked b? Remember, we need to have even outcomes that are not multiples of 7.

So, that should be 44, 46, and 48.

It's time now for your first task.

Jacob and Laura submit homework for showing the outcomes of two events from a trial using different representations.

Please use Jacob's correctly written outcome table to circle any errors in the outcome tree and write down what is wrong with the outcome tree.

Pause and do this now, please.

For question 2, this outcome tree correctly shows two events from the trial of spinning a spinner with eight possible outcomes.

Use the outcome tree to complete the Venn diagram.

Pause and do this now.

Welcome back.

Question 3, both this outcome tree and outcome table are showing the same two-stage trial: two spinners that will be spun, once each, with their outcomes summed together.

Both diagrams are accurate, but have some information missing.

The outcome table shows the sample space of outcomes after summation.

Please complete both diagrams. Pause and do this now.

Welcome back.

Let's go through our answers.

So, remember you had to circle what was wrong in our outcome tree.

Well, 1 was wrong because 1 is not prime, and the 5 is in the wrong place because 5 is a factor of 45.

Now, you may have noticed that 7 was missing, so you may have just made a note of that rather than circling anything.

7 is a prime number that is not a factor of 45.

So, in fact, that should be on there.

And the 21 needs to be removed, it's not even an outcome of this trial.

For question 2, you had to use the outcome tree to complete the Venn diagram.

Please pause now while you check that you have the same outcomes in the same regions as I do.

And let's mark question 3.

So, here you had to use the outcome table and the outcome tree to fill in each other to complete them.

So, here you had to use the partial information you were given to fill out everything.

In order to get started, I would've found the bottom row and found the header for that row.

So, for example, the 10 that I see in the bottom right cell, I know that the column heading is 3, and I know the answer is 10.

so what do I add to 3 to make 10? Well, I add 7, and that allowed me to fill in the 7 there.

Because I know that row has a 7 as its header and I know that there's an 8 in that row, I know that the corresponding column heading must be 1.

Once I've done that, I can look across to my outcome tree and see that paired with the 7, the alternative outcome is 4.

Well, that tells me that the other row must have a 4 as its header.

And then, because I've already filled in 1 and 3 as my column headers, I know the final column header must be -1.

At this point, I can fill everything in.

Well done if you've got this right.

It's now time for the second part of our lesson on using multiple diagrams. Jun says, "It can be very useful to have loads of different diagrams." And Sofia says, "But it can be tricky to choose.

When is one diagram better than another?" So, when do I want to use an outcome table, when would I like an outcome tree, what about a Venn diagram, when's that best? Alex points out, "Is it even possible to use every diagram for every situation?" Hmm, well, let's start thinking about these.

Sofia points out that lists are super straightforward if there are only a few outcomes.

So, for example, if I only want to consider the outcomes from spinner one, I could just list them, there's not a lot after all.

I agree with Sofia though, making a list is really tricky if there are lots of outcomes for each stage of a trial.

So, for example, if I'm gonna spin spinner one and spinner two, these are all different possible outcomes I can get, and there's a lot.

Look, I'd even missed one out, it was really hard to spot because these are listed not systematically.

And even if I did list them systematically, there's so many, it'd be easy to make an error.

Wow, that's what the outcome tree would look like.

Well, I guess it would.

There are six possible outcomes for spinner one and then five possible outcomes for spinner two.

So, yeah, I'd have all these branches, and that does look difficult to label.

I don't like that much either.

Sofia points out, "I know it's still possible to write down all outcomes using a list or an outcome tree," and she's right, "but maybe the outcome table will be better here." It's certainly more compact and because the columns and the rows or the headings for them are our outcomes, it's easy to make sure that I've listed every possible outcome I can get.

Alex points out, "The Venn diagrams are his favourite, but is it possible to draw one for these two spinners?" Ah, Jun's right, "A Venn diagram can only be used for specific question types." It wouldn't work here because what we've got here is A, B, C, D, E, F on one spinner and 1, 4, 8, 9, 15 on the other.

There's no overlap between these two.

Venn diagrams are very effective at categorising outcomes from a trial into events, and that's not what we had before.

For example, spinner two will be spun once.

What are the outcomes of the events that spinner will land on even or a square number? And now, we can see it's possible to say that one of our two circles will be to do with outcomes that are square and the other is to do with outcomes that are even.

Well, 4 is both square and even, 8 is just even, 1 and 9 are just square, and 15 is neither square nor even.

That was really easy to put into a Venn diagram.

However, Venn diagrams are not effective at systematically listing all the outcomes from a two-stage trial, and this is exactly what we were mentioning earlier.

And there's no way of systematically listing all the outcomes if spinner one and spinner two are spun once each.

For example, where would the outcome A8 go on this diagram? There's nowhere for it to go.

Let's do a quick check.

Jacob will spin the spinner and roll this fair six-sided dice with the integers from 1 to 6 on it.

Which of these can Jacob use to systematically write down all outcomes of this two stage trial? Remember, we're focusing on what he can use, not on what necessarily is best.

Remember, best can mean different things to different people.

Pause and make your choice now.

Welcome back.

Which ones did you pick? Well, a list, an outcome table, and an outcome tree would all work here.

Although, some of them might look quite complicated.

The one we absolutely cannot use, however, is a Venn diagram.

Remember, Venn diagrams are not helpful for systematically listing outcomes for a trial of two or more stages.

Jacob will spin this spinner and consider the events that the spinner lands on a negative number and the event that it'll land on an odd number.

So, which of these can Jacob use to systematically write down all outcomes for the two events? Pause and make your choice now.

Welcome back.

You could have used any of these 'cause all are helpful for categorising outcomes for different events.

It's now time for your final task.

For question 1, Jacob will spin this spinner and consider the events the spinner will land on a negative number and the event that it lands on an odd number.

For part a, please complete each diagram for the two events.

And then in part b, write a sentence explaining which diagram you prefer.

Remember, it's the diagram you prefer, not the one that a friend prefers or someone else you know prefers, it's which one you prefer.

And you do need to explain why you prefer it, please.

Pause the video and complete question 1 now.

Question 2, Sofia is going to choose a starter and a main course.

In part a, I'd like you to choose two different representations and write down the possible outcomes of Sofia's meal.

And then in part b, write a sentence explaining why you chose those two representations.

Pause the video and do this now.

Welcome back.

Let's go through our answers.

So, for part a, we had to complete each diagram for the two events.

Please feel free to pause at this point so that you can carefully check where I have my outcomes and make sure they align with yours.

Pause now.

In part b, you had to write a sentence to explain which diagram you prefer.

Now, this is personal to you.

You might have said that you like the outcome table because it's really concise and it's very easy to see all possible outcomes.

You might have said you like the outcome tree because it's easy to read along and you can make sure you have all possible outcomes.

You might have said the Venn diagram because you find it the most visual and easiest to see, especially when we're considering two events.

Remember, any answer here is correct as long as you've been very truthful about which one you prefer.

For question 2, I asked you to choose two different representations.

Now, you could have chosen to either list these or you could have put them into an outcome table or an outcome tree.

I've given you an example of an outcome table that represents this two-stage trial.

So, here you have our table, and I've chosen to put starters on the rows and main across for the columns.

Now, the outcomes you see here should either just be listed if you did a list, or if you've done a tree diagram then you could have done either the mains as your first layer of branches and starters as your second, or you could have done these the other way around.

Either would've been fine, but you should still reach the same eight outcomes.

Then for part b, we said to write a sentence explaining why you chose these two.

So, for example, if you chose a list, you might've said, "The number of possible combinations is small, and lists are very easy to write." For an outcome table or an outcome tree, you might've said that, "It makes it easy to see that you found all possible combinations." Remember, it's a sentence explaining why you chose the two representations.

So, it's absolutely fine for you to say something different to what I've said here on the screen.

It's now time to sum up what we've learned today.

It is possible to transfer information from one diagram that lists the outcomes of a trial or pair events to another diagram.

Lists, outcome tables, and outcome trees can all be used to systematically list all the possible outcomes of a two-stage trial.

Whilst these diagrams can all be used for any two-stage trial, one diagram may be preferred over another, dependent on the circumstances.

And lists, outcome tables, outcome trees, and Venn diagrams can all be used to categorise the outcomes of a trial into events.

Well done, you've done a great job today.

And I look forward to seeing you for more lessons on probability.