Lesson video

Hello there and welcome to today's lesson.

My name is Dr.

Rollinson and I'll be guiding you through it.

Let's get started.

Welcome to today's lesson from the unit of Probability and theoretical probabilities.

This lesson is called Checking and Listing Possible Outcomes.

And by the end of today's lesson, we'll be able to systematically list all the possible outcomes for one or more events.

Here are some previous keywords that we're gonna use again during today's lesson.

So you may wanna pause the video if you want to remind yourselves what any of these words mean before pressing play to continue.

This lesson contains two learning cycles.

In the first learning cycle, we'll be listing possible outcomes for one-stage and two-stage trials.

Then the second learning cycle, we're gonna do the same again, but for three-stage trials.

Let's start off with one-stage and two-stage trials.

Here we have Lucas.

And Lucas has a spinner.

The spinner has two sectors on it.

One says win and the other one says lose.

Lucas is going to spin the spinner once.

What are the possible outcomes when Lucas does this? Well, there are two possible outcomes.

Lucas can either win or he could lose, and we can represent these outcomes more formally as a sample space using this notation here, where W in this case represents win and L represents lose.

Now this is quite straightforward because there are only two possible outcomes on the spinner, and Lucas only spins the spinner once.

But what if Lucas spins the spinner twice? What are the possible outcomes this time? Maybe pause the video while you think about all the possible outcomes when Lucas spins the spinner twice and then press play when you think you've got them all.

I wonder what we thought, and in particular, I wonder what order you thought about those outcomes in.

Did you have some sort of system for what order you thought about them in? Or did you just think about the outcomes in whatever order they came to mind? Well, let's take a look at it together.

When Lucas spins the spinner twice, he could get any of these outcomes here.

He could win and then win, win and then lose, lose and then win, or lose and then lose.

And we could represent these four outcomes as a sample space like this as well.

Now, you may notice that these outcomes are not just listed in any random order.

They are listed in a specific order.

Can you spot what system has been used to order the outcomes? Lucas says, "To begin with, it lists all the possible outcomes where the first spin is a win." So we have win and then win and win and then lose.

And then once it's exhausted all the possible outcomes where the first spin is win, it then lists all the possible outcomes where the first spin is lose.

Lose and then win and lose and then lose.

And then with the second spin, the second spin alternates between win and lose each time.

Now Laura says she has a different way to explain a system of listing the possible outcomes of the two spins.

Laura says, "I first listed the outcomes for the first spin," and she's listed them as an outcome tree where one possibility is win or the other possibility is lose.

So that is what's going on with the first spin.

And then she says, "For each outcome, I then list all the possible outcomes for the second spin." So we can see if the first spin was win, then there are two possible outcomes for the second spin, win or lose.

And if the first spin was lose, there were two possible outcomes for the second spin again, win and then lose.

And then she says, "I then followed my finger along each root of branches to create my list." So top root goes win and then win, the next one underneath goes win and then lose and then lose, and then win, and then lose and then lose.

So we have two different ways of thinking about that system for ordering those outcomes.

Now you don't have to always follow that particular system for ordering outcomes.

You can use a different system, but having some kind of system in mind can be helpful to ensure that you don't either repeat an outcome twice or miss out an outcome.

Lucas says, however, "This system for list outcomes could get quite tricky when listing lots of outcomes." And we'll see plenty of examples of those during today's lesson.

So let's check what we've learned.

Alex is using a system to list all the possible outcomes when the spinner below is spun twice.

So far he's listed AA, AB, AC, AD, BA, BB, BC, BD.

And what would the next outcome be using Alex's system? Pause the video while you write it down and press play when you're ready for an answer.

Well, he has listed all the possible outcomes for the first spin being A, and then all the possible outcomes for the first spin being B.

So the next one, the first spin will be a C.

And you can see the second spin, it's alternating between A, B, C, D and A, B, C, D.

So his next one is gonna be C and then A.

Did we get it right? Well done if you did.

A coin is flipped and a regular, six-sided dice is rolled.

So one possible outcome could be that you get a heads on the coin and you roll the number one, but what are all the possible outcomes? List down all the possible outcomes for when this event happens.

Pause video while you do that and press play when you're ready to continue.

The possible outcomes are these: you can get heads in a one, heads in a two, heads in a three, and then up to six.

Or you can get tails and one and tails and two and so on, up to tails and six.

Okay, it's over to you for Task A.

This task contains two questions, and here is question one: There are two decks of cards.

We have deck A, which has cards numbered from zero to two, and then we have deck B, which has the cards listed from zero to nine.

And you've got four questions to answer regarding these two decks of cards.

Pause the video while you have a go at this and then press play when you're ready for question two.

And here is question two.

Here is a restaurant menu for two courses.

A customer chooses one starter and one main course, and you need to list all the possible outcomes for a customer's meal.

And then describe the system that you used with a list and all the outcomes in A.

maybe write a sentence or two to try and describe your system.

Pause the video while have a go at this, and then press play when you're ready for some answers.

Okay, well done with that.

Let's now go through some answers.

So in part A, we have to list all the possible outcomes for drawing a card from deck A.

Well, here they are.

You can get a zero, a one or a two.

And then part B, you had to list all the possible outcomes when drawing a card from deck B.

Well, here these are, you can get a zero, a one, a two, a three, a four, a five, a six, seven, an eight, or a nine.

And in part C, you had to list all the possible outcomes when drawing one card from each deck.

So what are those? Well, here they are.

And then in part D, we need to describe the system that we used to list all the possible outcomes in part C.

Well, in the way that it's done on the answers here, it's kind of like how we count.

So we start off with a zero from deck A and a zero from deck B, so zero zero.

And then if we keep the first digit the same to begin with, the first card the same and increased the value of the second card, the one from deck B each time until they've all been used.

So 00, 01, 02, and so on.

And then once we've exhausted all those possible outcomes, we then increase the value from deck A by one and repeat the same process just like how we count from one to 29 or from zero to 29 in this case.

Question two: here is a restaurant menu for two courses and we have to list all the possible outcomes from when a customer chooses a meal with one from each course.

Well, we could list them it in different ways, but we should have these outcomes here.

Now, you may have listed them in the same order as I've done here.

You may have used a different order, but what you should have still is 12 different outcomes where for four of them they have soup.

And for four of them there's calamari.

And for four of them there are meatballs.

And then also you should see lasagna appear three times in the outcomes and ravioli and bolognese and pizza as well.

And then in part B, we need to describe the system that we use to list the outcomes in part A.

Now, if you used a different system to me, you all might write something a bit different here, but let's describe the system that we've used in these answers.

We started with a top item from each course, soup and lasagna, and then we kept the first item the same and worked through the main course items in order until we reached the end of the list.

And once we did that, we moved to the next item in the starter and worked through this, this main courses again in the same way.

And we repeated the same process again and again until the final starter had been paired with every single main course.

Well done so far.

Let's now move on to the second learning cycle where we're gonna do the same thing again, but this time it's more complex 'cause we've gotta have three-stage trials instead.

Here we have Aisha, and Aisha has the win-lose spinner that we saw earlier.

Now the possible outcomes for spinning the spinner twice are listed below.

Win-win, win-lose, lose-win and lose-lose.

How could these be used to list all the possible outcomes when the spinner is spun three times? So rather than start from scratch and try and list all the possible outcomes when the spinner is spun three times, let's look at how we could adapt the ones we have already for two spins to make it so it's three spins.

Pause the video while you think about how you might achieve this and press play when you think you've got an idea in mind.

I wonder what we thought.

Let's see what Aisha thinks.

Aisha says, "For each of the current outcomes, there are two outcomes for what the additional spin could be." So for example, if we take a look at the top outcome on that list, the first spin is win, the second spin is win, and we're about to spin the spinner for a third time.

Well, we could either get a win or a lose on that third spin.

So there'll be two outcomes where the first two spins are win, and that's the case for each of the outcomes we've listed so far.

Aisha says, "I could copy each of the outcomes I have so far and then write win after one of the outcomes and lose after its copy." So for example, in the top row we have win-win-win, and we have win-win-lose.

Those two outcomes are the same for the first two spins and only differ with the third spin and the same as well for the next row.

Win-lose-win and win-lose-lose.

They are the same for the first two spins and differ for the third spin.

Now here, we've considered the additional spin that we're writing to be the third spin, but Aisha says, "I can consider the first spin to be the additional spin and so write the extra outcomes at the start of each." So we have all the same outcomes here, but they're in a different order.

Here we can see that the left column has all the outcomes where the first spin is win, and the right hand column has all the outcomes where the first spin is lose.

Here we have Jun, and Jun has a spinner with letters A, B, and C on, and Jun is going to spin the spinner three times and he wants to list all the possible outcomes.

Now he's spinning it three times and each time he spins it, there are three possible outcomes.

So this could be quite a lot of different outcomes here.

And if we just try and think about them in whichever order comes to mind, there's a good chance where you might miss some outcomes out or maybe repeat the same outcome twice.

So a system will be very helpful for Jun in this situation.

So how could Jun do this systematically? Pause the video while you think about what system Jun might use to list all the possible outcomes.

And then press play when you think you've got an idea.

I wonder what system you thought about here.

Now there are lots of different systems you could use and there's no right or wrong way of doing it, but let's see what Jun thought.

Jun says, "I could start with A-A-A." So all three spins are A, "I then could cycle through the different possibilities for the third spin from A to B to C." So we've got three outcomes so far, and all three of the outcomes, the first two spins are A, and the third spin differs.

A, B, C.

Jun then says, "I could then copy all three of these outcomes and cycle through the different possibilities for the second spin: from A to B to C." So for these three outcomes, we can see the second spin is A.

If we copy them and change the second spin to B and copy all three again and change them to C, we have three sets of outcomes here where the only thing that differs is the second spin.

In the first set of three, it's A, then it's B, and then it's C.

Jun then says, "I could copy all of these outcomes I have so far and cycle through the different possibilities for the first spin." So here we can see the first spin is A, if we copy them all over, but change the first spin to B and then copy them all again and then the first spin is C, we now have all the possible outcomes for if this spinner is spun three times.

So let's check what we've learned there.

Here we have a coin, a regular six-sided dice and a spinner with the letters A, B, and C on.

The coin is flipped, the dice is rolled and the spinner is spun.

So one possible outcome here is you could get a heads on the coin, the number one on the dice and an A on the spinner.

Now that's one outcome, but there are actually 36 different possible outcomes that could happen here.

We've made a start in listing a bunch of them, but we have three outcomes left to list.

Now I wonder if you can spot which outcomes are missing and what might help you is if you can spot what system has been used so far to list the outcomes we have already.

Pause the video while you read through what's going on here.

Try and spot the system and write the three remaining possible outcomes.

Then press play when you are ready for an answer.

Okay, before we reveal the answer to that, let's try and crack this system what's been used so far.

If we look at how these outcomes are laid out, we have the top two rows have heads in, and the bottom two rows all have tails in.

So those three blank spaces on that bottom row, they'll all have tails.

And then if we look at also, these outcomes are grouped in threes.

We can see the first three all have the number one in, the second three all have the number two in, and then the number three and then the number four and so on.

So we can see that each set of three outcomes cycles between the different numbers from one to six.

Now the last set of outcomes before the blanks all have five in.

So these last three will have six in.

And then in terms of letters, if we look vertically down the list of outcomes, we can see the vertically, the four outcomes on the left all have A in.

And then the next four, all of B in, and then C, and then A, then B, then C.

So we're cycling between A, B, and C each time we move from left to right.

So those last three outcomes will have A and then B, and then C.

So here we are.

Tails six A, tails six B, and tails six C.

Did you get it? Well done if you did.

And then here we have the same scenario again, but the outcomes have been listed using a different system.

I wonder if you can spot the system this time and figure out what are the six remaining outcomes that need to be listed.

Pause the video while you do this and press play when you're ready for an answer.

Okay, once again, before we reveal the answers, let's try and crack the system.

Let's see where heads and tails appear in the system.

What we can see is all the ones on the left half of the list all have heads in and all the outcomes on the right half of the list all have tails in.

Now these six remaining outcomes are all on the right-hand side of the list.

So they'll all have tails in.

And what's going on with the numbers? Well, if we look vertically, we can see that on the far left, those three outcomes that are all vertically above each other, they all have one in.

And the next three outcomes on the right all have two in, and then three and four and five and six and so on.

So as we move from left to right, the numbers are cycling between one and six.

So then the last outcome we can see has a six in it.

The next one must be one, then two and three, then four and five and six.

And then what about the letters? Well, the top row all have As in, the second row all have Bs in.

And the third row, they will all have Cs in.

So our six remaining outcomes are these: tails one C, tails two C, tails three C, tails four C tails five C, and tails six C.

How did you get on with that? Well, it's over to you now for Task B.

This task contains three questions, and here are questions one and two.

In question one: A flipped coin has two outcomes, heads and tails, and you need to complete the list for all the possible outcomes when a coin is flipped three times.

In question two, a netball team plays three matches.

In each match, they could either win, lose, or draw.

Can you list all the possible outcomes for results for these three matches? Now, for both of these questions, it'll be helpful to have a system that you followed and think about some of the systems you've seen here today.

Pause the video while you have a go at these and then press play when you're ready for question three.

And here is question three.

We have the restaurant menu again, but you may notice for this time there's a dessert menu.

Yummy.

This time the customer chooses one starter, one main, and one dessert, and you need to list all the possible outcomes for a customer's meal.

Now, you may use your answers from Task A to help you if you have them to hand.

Otherwise, follow a system and list all the outcomes from scratch.

Pause the video while you do this and press play when you're ready for some answers.

Okay, let's now go through some answers.

Here is question one: A coin is flipped three times, and we need to list all the possible outcomes.

So far we have heads, heads, heads, heads, heads, tails.

And if we continue the same system, we'll have these outcomes here.

And in question two, a netball team plays three matches.

Each time they could either win, lose, or draw.

We need to list all the possible outcomes from three matches.

Well, here they are.

Now you may have listed your outcomes in the same order as these have been listed, but if not, you should still have the same number of outcomes.

You should still have nine lots of three outcomes, 27 altogether, and you should have the same outcomes.

And here is question three: we're to list out all the possible outcomes when choosing a starter, main and dessert from this menu.

And here they are.

You should have 24 outcomes where eight of them have soup, eight of them have calamari, and eight have meatballs.

Also, you should see as well that six of your outcomes have lasagna, six have ravioli, six have bolognese, and six have pizza.

And in terms of your desserts, well, you should have 12 outcomes with ice cream and 12 outcomes with tiramisu.

Great work today.

Let's now summarise what we've learned in this lesson.

The possible outcomes for a single stage trial can be listed.

They're usually quite straightforward, but also the possible outcomes for a trial with multiple stages can also be listed.

A two-stage trial, such as flipping a coin twice or a three-stage trial, like flipping a coin three times.

But the more trials we have or the more stages to a trial we have, the more complicated it gets to list all the possible outcomes.

So having a system for how you order the outcomes can be helpful when listing outcomes for trials with multiple stages.

And different systems can be used when listing outcomes.

There's no one set system.

You can do it in whatever way you like, so long as you have some kind of ordering mechanism in mind.

Well done today.

Thank you very much.