Lesson video

Hello there, and thank you for choosing this lesson.

My name is Dr.

Rowlandson and I'll be guiding you through it.

So let's get started.

Welcome to today's lesson from the Unit of Conditional Probability.

This lesson is called Checking and Securing Displaying Outcomes, and by the end of today's lesson, we'll be able to display outcomes in various ways and place outcomes on a probability scale.

Here are some previous keywords that will be useful during today's lesson.

We will recap what they mean during the lesson, but you may want to pause the video if you want to remind yourself what they mean before we start and press play when you're ready to continue.

The lesson is broken into two learning cycles.

In the first learning cycle, we're going to be focusing on how to list outcomes in a systematic way.

And then the second learning cycle, we're going to look at how to write probabilities of these outcomes and place them on a probability scale.

Let's start off with listing outcomes.

Before we get into today's lesson, let's look at some of the key words and notation that we're going to be using quite a lot during the lesson.

A trial is a single predefined test.

For example, a trial may have a single stage such as spinning a spinner once.

But a trial may also have multiple stages to it, such as flipping a coin three times.

The whole thing will be considered as a trial.

A trial may also be other things, such as people running in a race.

The outcome is a result of a trial.

Now, the outcome could be a numerical outcome such as which number you roll on a dice or how many times something happens in a certain space of time.

Or an outcome could be a worded or descriptive outcome instead, such as which person wins the race, the name of the person who wins the race for example.

A sample space is the set of all possible outcomes for a trial.

And an event is a subset of a sample space.

It could be a particular outcome or it could be a set of outcomes that may occur from a trial.

And the set of all possible outcomes of a trial can be written using the following notation.

We've got our Greek letter xi, and that represents the universal set.

And then we have our curly brackets with the outcomes written in between those curly brackets, and that will be our list of all the possible outcomes, our sample space.

Let's take a look at an example of that to illustrate what all these things mean.

Here we have Izzy, who's going to spin this spinner once.

So the trial in this case is one spin of the spinner, and then if we use formal notation to write down all the outcomes for this, we would have our symbol xi, which represents the universal set, the set of all possible outcomes that is.

And we have equals.

And then in the curly brackets we have all the possible outcomes, the letters A to H, so those are our outcomes.

The full set of outcomes is our sample space.

And then what would events be? Well, events could be whatever it is you define the events to be.

Well, in this case, we could say that one event is getting a vowel and another event is getting a consonant.

Now, these events have different outcomes.

To get a vowel, you could get A or E on this spinner.

To get a consonant, you could get B, C, D, F, G or H on it.

So we have our outcomes, our sample space, and our events.

Let's look at another example of this.

I'm gonna do one on the left and then you'll go try one very similar on the right.

Here we have a coin, which is flipped once.

So our trial is one flip of the coin.

Let's write down a sample space to show all the possible outcomes.

So here it is, I've got my symbol, xi, and it's equals, and we've got our sample space of all possible outcomes, heads and tails.

So here's one for you to try.

We have a regular six-sided dice and it's rolled once.

Could you please pause the video and then write down the sample space to show all possible outcomes using the notation, and press play when you're ready to see an answer.

Let's take a look at the answer.

Hopefully you've got something that looks a bit like this.

Greek letter xi, which represents the universal sets, and that's equal to the numbers 1, 2, 3, 4, 5, and 6 in our curly brackets.

Here's another question for you to try yourself.

A trial consists of one spin of a spinner below.

Could you please write down a sample space for this trial? Pause the video while you do it and press play when you're ready for an answer.

Let's take a look.

Hopefully you've got something a bit like this, down left, right, and up.

Now you might have those outcomes in a different order, and that's absolutely fine.

Here's another one.

We've got a spinner and the sample space shows the outcomes for a single spin of this spinner.

However, you can see there's a blank on the spinner.

Could you please fill in the blank for that spinner based on what it says in the sample space? Pause while you do it and press play when you're ready for an answer.

The blank would be skip.

That is the only thing that is missing from that spinner that is in the sample space.

Here we have a spinner with three sectors with letters A, B, C, and D on it, and we have Alex.

Alex lists all the possible outcomes for one spin of this spinner.

He says, "The outcomes for one spin are either A, B, or C." And we could write that with full notation like this.

Now in this case, the trial is one spin of the spinner, but Alex is going to change that.

Alex is going to spin the spinner twice.

So how could he list all the possible outcomes for the two spins? Pause the video while you think about this, and in particular, think about how could you order those outcomes in a systematic way so that you don't either miss any of the outcomes out or repeat any of the outcomes.

Once you've thought about it, press play when you're ready to continue.

Well, rather than starting from the beginning, Alex says, "I could adapt my list for one spin to produce a list for two spins." Let's see how he does that.

Alex lists all the possible outcomes for two spins of the spinner.

Here we can see the outcomes for one spin, but Alex says, "I could use three copies of the list for one spin like this, and that would be one copy for each outcome of the additional spin." So you could view this as being what we have on the screen at the moment, other possible outcomes for the second spin.

And what Alex is about to do now is write down what happens in the first spin.

He says, "The first copy are all the possible outcomes for if the first spin is A." So we have AA, AB, and AC.

And they're written as well with our formal notation at the bottom.

The second copy are the possible outcomes for if the first spin is B, so you have BA, BB, and BC.

And the third copy are all the possible outcomes for if the first spin is C, CA, CB and CC.

So now we have all the possible outcomes for two spins of the spinner.

Let's change it again.

How could Alex adapt this sample space to list all the possible outcomes for three spins of the spinner? Pause the video while you think about this and press play when you are ready to continue.

Well, once again, we don't have to start from the very beginning.

Here we have all the possible outcomes for two spins.

Alex says, ""I could use three copies of this list, and that's one copy for each outcome of the additional spin." So here we have all the possible outcomes for two spins of the spinner.

If we consider those as being the last two of the three spins, then it means if we make three copies of this, we can fill in what the first spin could be.

Alex says, "The first group shows all the possible outcomes for if the first spin lands on A." So every single one of those outcomes has a followed by every possible outcome for the last two spins.

The second group is for if the first spin lands on B.

So we have B followed by every possible combination of outcomes for the last two spins.

And then the third group is for if the first spin lands on C.

We have C followed by every possible outcome for the last two spins as well.

So now we have all the possible outcomes for three spins of the spinner written in a systematic way.

Now, you didn't have to list them in the same way as Alex.

Outcomes can be listed in any order, but it can be helpful to use a systematic approach to listing outcomes.

It could be the same way as Alex, or it could be anyway, so long as there's some kind of order that you can keep track of so you know you don't miss any outcomes out and you know you haven't repeat any outcomes as well.

So let's review the systematic approach that was taken with this particular spinner.

The spinner has three sectors, so each time you spin it, there are three possible outcomes.

So what we have on the screen here are all the possible outcomes for one spin.

Now, for each time you spin it, there are three possible outcomes.

So if we have two spins, then we could write three copies of the outcomes of one spin, and then write all the possible outcomes for the other spin like so.

And then to adapt this to three spins, do the same process again.

Make a copy for every single possible outcome of the additional spin and pair every single possible outcome for two spins with one of the outcomes for the additional spin like this.

And now you can be sure you have all the possible outcomes for three spins.

And if you was to do all the possible outcomes for four spins, you do the same thing again.

You make three copies of this and then pair each one of them up with A, B or C.

And then for five spin, you make three copies of that and so on.

So let's check what we've learned.

Here we have a spinner which is spun once.

Write down a sample space for this particular trial.

Pause while you do it and press play when you're ready for an answer.

Let's take a look at the answer.

You could write your answer as a list, win and lose.

Or you can write it more formally using the notation we have, xi equals and our curly brackets, we have W and L.

Now that shows that win and lose have been abbreviated to W and L.

That's absolutely fine to do.

But if you do do that, it is helpful to write a key somewhere or a sentence that tells you what each of those letters stand for.

So how about if the spinner is spun twice? Could you please pause a video and write down a sample space for this trial and then press play When you're ready to see the answer.

Here's our answer.

You should have four possible outcomes, win-win, win-lose, lose-win, and lose-lose.

Or you can write it with a formal notation with the words abbreviated.

Now, you may have written them in the same orders that you can see here, or you may have written them in a different order.

How about if we spin the spinner three times? Write down the sample space for this trial.

Pause while you do it and press play When you are ready for an answer.

Let's take a look at the answer.

You can see the answer displayed as a list, win-win-win, win-win-lose, and so on, and you can see it displayed as formal notation.

Now once again, you may have written it in the same order as you can see here, or you may have written it in a different order.

If you've written a different order, check, but you should have eight possible outcomes.

And for example, four of those outcomes would have win as a first spin, four would have lose as the first spin.

And you should have four outcomes with win in the second spin, and four will lose in the second spin.

Four outcomes with win in the third spin, and four with lose in the third spin.

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

This task contains three questions, and here is question one.

Pause the video while you do it and press play when you're ready for question two.

Here is question two.

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

And here is question three.

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

Okay, let's take a look at some answers.

For question one, you had two decks of cards.

Deck A had letters A to D and Deck B had numbers one to five.

For part A, all the possible outcomes for drawing a card from Deck A would be A, B, C, and D.

For part B, all the possible outcomes for drawing a card from Deck B would be 1, 2, 3, 4, and five.

And then for part C, one card is drawn from each deck and you have to write down every single possible outcome.

Well, your key job here is to pair every outcome from Deck A with every outcome from Deck B and do it in a way that is systematic so you ensure you don't repeat anything or miss anything else.

In this approach, the thinking behind it is to pick each outcome from Deck A and pair it with each outcome from Deck B.

So for example, if we get A, what numbers can we get with A, A1, A2, A3, A4, A5, and then if we get B from Deck A, what numbers can go with B and what numbers can go with C and what numbers can go with D? You may have approached it a different way, you may have thought about if I get the number one from Deck A, what letters could go with that? You might have gone A1, B1, C1, D1, and then A2, B2, C2 and D2 and so on, or you may have used a different ordering.

The important thing is you should still have 20 outcomes no matter what order you've written them in.

Five of those outcomes should be for each letter and four of those outcomes should be for each number.

Then question two, Sam and Jun are playing a game of rock, paper and scissor.

and in part A, you have to complete a table to list all the possible outcomes of the game.

Well, your table could look something a bit like this.

You may have written 'em in a different order and that'll be absolutely fine.

You should still have nine possible outcomes altogether, there are three possible outcomes for Sam and for each of those outcomes, there are three possible outcomes for Jun.

And part B, how many outcomes satisfy the event that both people show their hand in the same way as each other? That'll be three possible outcomes.

Rock-rock, paper-paper and scissor-scissors.

How many possible outcomes satisfy the event that they show their hand in a different way to each other? That'll be six.

And then how many outcomes satisfy the event that at least one person shows paper, that would be five.

Then question three, a restaurant has a three course menu.

A customer chooses one dish from each course, and in part A, you have to list all the possible outcomes for a customer's choice of three courses.

Let's take a look at one possible way you could have written your answer.

Here's one, you may have written your answer in the same order as these, or you may have used a different order and that's absolutely fine as long as you have them all and you haven't repeated any.

Let's take a look at the ordering that was used here.

The first outcome in this sample space is SBI.

That's the top item from each of those courses.

Let's take a look at what happens as we go through this sample space.

They started with a top item from each course and then looked at what happens when we move one item down each time with desserts, SBI, SBJ.

And then now we've got to the bottom of the dessert list.

We can move one place down in the main list, so we have SHI and SHJ.

And then move one place down the main list again, SCI and SCJ.

And now we've got to the bottom of the main list, we can move the starters one place down, PBI, PBJ, and then keep repeating in the same way.

And what we can see then is that there should be 12 possible outcomes altogether.

In part B, it says, how many of these satisfied the event that our customer chooses soup? That'll be six.

And in part C, how many outcome satisfy the event that a customer chooses beef? That would be four.

You're doing great so far.

Now let's move on to the next part of this lesson where we look at probabilities and the probability scale.

Let's remind ourselves what probabilities are.

Probability is a numerical measure of the likelihood that an outcome happens.

Now, often with probabilities, words such as likely and unlikely are used to describe the likelihood that an event will occur.

But the problem of these words is that they are relatively vague.

For example, two events can both be described as being likely to happen, but one of those events might be more likely than the other.

So the words alone don't really give us much sense of how much more likely one thing is than another.

Probabilities represent the likelihood of an event in a more precise way by using a numerical value to indicate its likelihood.

And the probability that an event will occur is the proportion of times the event is expected to happen in a suitably large number of trials.

Here we have a scale with likelihoods from impossible to certain.

In the centre of that scale, we have the words even chance.

That means that something is just as likely to happen as it is to not happen.

Above even chance, we have things that could be described as likely.

Those are outcomes that have a greater chance of happening than not happening.

And then below even chance, we have events or outcomes that could be described as unlikely to happen.

Unlikely meaning that there's a greater chance that it doesn't happen than there is, but it does happen.

And we can turn this into a probability scale by placing numerical values along the scale.

An event that is certain has 100% probability of happening.

For example, with this spinner, it only has one possible outcome which is win.

So if you spin it, no matter how many times you spin it, you'd expect 100% of the trials to be win.

So the probability is 100%.

An event that is impossible has a 0% probability of happening.

For example, with this spinner, no matter how many times you spin it, you'd expect zero of the trials to be win.

So getting a win on this spinner would be 0% probability.

An event that has an even chance has a 50% probability of happening.

For example, with this spinner, if we conducted a large number of trials, then we'd expect that approximately 50% of the outcomes would be win, so we could say the probability of getting a win with this spinner is 50%.

Probabilities that are greater than 50% have more than an even chance of happening.

For example, with this spinner, if we conducted a large number of trials, then we would expect it to land on win more often than we would expect it to land on lose, so its probability will be greater than 50%.

For example, it might be 75%.

Probabilities that are less than 50% have a less than even chance of happening.

For example, with this spinner, if we conducted a large number of trials, then we would expect it to land on win less often than it lands on lose, so we'd say that probability is less than 50%.

It might be something like 25%, it'd be unlikely to get a win on that spinner.

Probabilities can also be expressed using equivalent decimals, so rather than going from 0% to a hundred percent, we can go from zero to one.

Zero means an event is impossible, one means event is certain, and the size of the number indicates how likely it is for an event to occur.

Probability can also be expressed using equivalent fractions, from zero to one with any equivalent fractions in between.

Sometimes it can be helpful to write them always with the same denominator, We don't necessarily have to do that.

For example, in this probability scale here, you can see all the fractions have been simplified.

And just like with decimals and percentages, the size of number indicates how likely it is for an event to occur.

Let's check what we've learned.

Which likelihood describes an event with a probability of one half? Pause the video while you choose and press play when you're ready for an answer.

The answer is even chance, one half is exactly midway between zero and one, which means you have just the same chance of something happening as not happening.

It's even chance.

Which likelihood describes an event with a probability of 3/8? Pause while you choose and press play when you're ready for an answer.

The answer is B, unlikely.

3/8 is less than half, which means it's less than an even chance.

There's a greater chance that this event doesn't happen than there is that it does happen, so it's unlikely.

The probabilities for three unrelated events are listed below, and these are the events, A, B, and C.

Which event is least likely to happen? Pause the video while you write it down and press play when you're ready for an answer.

The answer is A.

The probability that A happens is 2/7.

And out of all those probabilities, that fraction has the lowest value, so it has the lowest probability and is least likely to happen.

Here we have three more unrelated events, D, E, and F.

Which event is least likely to happen here? Pause the video while you choose and press play when you're ready for an answer.

The answer is F.

The probability that F happens is 4/9.

Now of those three fractions, that one has the least value, which means is least likely to happen.

This question is different to the previous one in that in the previous question you had the same denominator and you had to compare the numerators.

In this one, you've got the same numerator and you have to compare the denominators.

If you wanted to, you could have made them all have a common denominator and compare the numerators.

Here you've got a spinner with win and lose on and you've got a probability scale, which arrow shows a probability that the spinner lands on win.

Pause while you choose and press play when you're ready for an answer.

The answer is B.

With this spinner, you can see the sector for win is slightly smaller than a sector for lose, so if you did a large number of trials, you expect to lose more often than win, which means it's less than a half chance of getting a win.

And it's only slightly less, that's why it's B and not A.

Statements about probability can be expressed more efficiently using a notation P, and in bracket you have event and equals.

For example, here we have a sentence that says the probability of a spinner landing on A is 3/5.

That sentence can be written in a much more efficient way using the notation you can see on the screen here.

The P stands for probability.

In the brackets we have the event we're talking about.

The event that the spinner lands on A.

And then we have the numerical value of the probability, 3/5.

For example, here we have a spinner with A and B on, and we have the notation that says P of A.

That means the probability the spinner lands on A is equal to one half.

Let's now think about how we get that half, how we'll write down the probabilities.

Well, the probability is one half because one out of the two possible outcomes are A.

If you look at the spinner, you can see those sectors are the same size, which means they have an equal chance of being landed on.

So there are two possible outcomes that are equally likely to happen, and one of them is A, so that's why it is one half.

And while you can write probabilities as decimals or percentages, fractions are particularly helpful because you can see very clearly what the numbers relate to.

The numerator shows us the number of outcomes that are A, there is one A in this spinner.

The denominator shows us a total number of outcomes altogether.

There are two possible outcomes for this spinner or two possible sectors that it could land on.

Now, the fact that probability is one half in this case means that in an experiment with a large number of trials, we'd expect approximately half of the outcomes to be A.

Now, it wouldn't necessarily be exactly half if we do it in practise, but we'd expect approximately a half of them to be A.

And it's based on the theory that it will land on each of the sectors an equal number of times.

However, this might not actually happen.

It's just an expected approximation.

Here's a different one.

We have a spinning out with five sectors, and the probability that this spinner lands on A is 3/5.

Let's consider where that 3/5 has come from.

This is because three out of the five possible outcomes are A, and each of those outcomes is equally likely to happen.

So we can see with our notation, we have P of A is 3/5 and the number of outcomes that are A is three, that's our numerator.

And the top number outcomes is five, that's our denominator.

So this probability means that if the spinner was spun, let's say 500 times, we'd expect it to land on each sector 100 times.

And then we'd expect it to land on A 300 times altogether.

300 is 3/5 of 500.

However, this might not actually happen.

It's just an expected approximation.

So let's check what we've learned.

Here you've got a spinner.

What does the probability but the spinner lands on the letter B? Pause while you write it down and press play when you're ready for an answer.

The answer is 1/5.

The five comes from the fact there are five possible outcomes or five sectors to the spinner.

And the one relates to the fact that only one of those sectors is B.

What's the probability that the spinner lands on a vowel? Pause the video while you write it down and press play when you're ready for an answer.

Well, the probability lands on a vowel is 2/5.

Once again, it's fifths because there are five sectors to this spinner.

It's two because two of those sectors have vowels.

That's A and E.

Here we have a regular six-sided dice, which is rolled once.

What's the probability when it lands on a multiple of three? Pause the video while you write it down and press play when you're ready for an answer.

The probability of rolling a multiple of three is 2/6.

Two relates to the fact there are two numbers that are multiples of three, that's three and six.

And the six relates to the fact there are six possible outcomes.

Now, you can simplify that to one third or you can use any equivalent fraction, decimal, or percentage.

If the dice is rolled 1,200 times, which of the possible options seems the most reasonable approximation for how many times the dice would be expected to land on a multiple of three? Pause the video while you choose and press play when you're ready for an answer.

The answer is B, 400.

400 is one third of 1,200.

Okay, its over to you now for Task B.

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

Pause the video while you do these and press play when you're ready for question three.

And here is question three.

Pause while you do this and press play when you're ready for question four.

And here is question four.

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

Okay, let's take a look at some answers.

In question one, a regular coin has two sides and it's flipped once.

The probability it lands and heads is one half.

If it's flipped a thousand times, you'd expect it to land on heads approximately 500 times.

And will the coin lands ahead exactly that many times? Probably not.

The approximation assumes the coin will land on each side an equal number of times, but it may not actually happen in practise.

You can try that out for yourself.

You could flip a coin, let's say 10 times and see if you get the exact same number of heads and tails.

You might do, but you might not.

Then question two, you have a bag of cubes, which are all the same size, some are blue and some are green.

In part A, a cube is chosen at random.

The probability it's green is 3/8.

There are three green cubes and there are eight cubes altogether in the bag.

Then part B says, would your answer to part A still be true if all the green cubes were larger than the blue cubes? No, the probability is based on a theory that all individual cubes are equally likely to be chosen.

If the green cubes are larger, it may affect the likelihood that they are chosen.

Then in question three, a game involves a set of cards, which you can see on the screen, and part A, you have to find the probabilities of particular events when one card is chosen at random from the set.

The probability drawing a rhombus would be 8/24.

because there are eight cards with a rhombus on and there are 24 cards altogether.

We can write that as a third or anything that is equivalent.

The probability of drawing a card that contains one shape would be 6/24, which is a quarter or anything equivalent.

And the probability of drawing a card with a hollow shape would be 12/24 or one half.

Then part B, you have to sort the events from part A in order of likelihood.

Where the least likely to happen is to draw a card containing one shape, followed by drawing a card containing rhombus, followed by drawing a card containing a hollow shape.

Then question four, you've got the five children playing a game which involves rolling a regular six-sided dice, and you have to write down the probability that each person rolls what they need to do to score a point.

Well, for Aisha, it'd be 1/6 because there's one three in the dice and there are six numbers altogether.

For Izzy, the probability rolling an odd number will be 3/6 because there are three odd numbers out of six possible numbers.

For Jacob, the probability of rolling the cube number will be 1/6.

There's one cube number on the dice, which is the number one.

For Sofia, the probability of rolling a factor of six will be 4/6 because there are four numbers in the dice which are factors of six, that's 1, 2, 3, and six.

And for Lucas, the probability of rolling a multiple of three is 2/6 Because there are two multiples of three in the dice, that's three and six.

In part B, you have to represent those likelihoods on the probability scale.

It would look something a bit like this.

Fantastic work today.

Now let's summarise what we've learned.

Outcomes can be listed systematically using an appropriate representation.

Outcomes can be placed on a probability scale.

And probabilities tell us the likelihood that the event happens from zero, which means the outcome is impossible, to one, which means the outcome is certain.

Well done.

Have a great day.