Lesson video

Hi, I'm Mrs. Wheelhouse.

Welcome to today's lesson on Equally Likely Outcomes from the unit Probability: Possible Outcomes.

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 state when outcomes will be equally likely.

Let's get going.

So, here is some keywords that we're gonna be using today in our lesson.

We've got trial, which is a single predefined test.

An outcome, which is a result of a trial.

A sample space, which is all the possible things or outcomes that can happen in a trial.

Now, we use something called a sample space diagram, which is a way of producing a sample space, allowing us to see it really clearly, and we do this in a systematic way.

And likelihood describes the chance that a particular event will occur.

So, you've probably talked about likelihoods loads in the past and just not realised it.

In fact, it's likely that you did that.

That's an example of a likelihood, and we'll explore that more in today's lesson.

So, our lesson today is broken into three parts, and we'll start with part one, listing possible outcomes.

So, this lesson's gonna look at scenarios where there are different possible things that can happen.

Now, these scenarios can be discussed using the following terminology, so the following words.

Remember a trial, we talked about this on our keyword slide, is a single predefined test.

So, for example, you could think of a trial as being spinning a spinner once.

You could think of a trial as being I flip a coin three times.

You could think of a trial as being a race between several runners.

An outcome is the result of a trial.

Now, it's just a result, it's not all the results.

An outcome is just a result of a trial.

And a sample space is the set or the group of all possible outcomes for that trial.

Now, the set of all possible outcomes can be written using the following notation.

Can you see that there? It effectively looks a lot like a squiggly E.

It's kind of the way I think of it.

And you can see there that we are using set notation, so that's the curly brackets, to show all the things that belong to the set of possible outcomes.

And we refer to that as our sample space.

Now, that squiggly E we talked about, that's the Greek letter xi.

You can see it there.

So, we write that as xi.

And it's used generally as a symbol to represent the universal set.

So, in the context of our topic of probability, we use it to represent the set of all possible outcomes for a trial.

So, let's have a look here at Jacob, who's gonna be spinning this spinner once.

What are the possible outcomes? In other words, what could happen when Jacob spins the spinner? What do you think? Well, let's think about this in terms of our terminology.

The trial is that Jacob is going to spin the spinner once, so we saw that in the question.

So, a trial is spinning the spinner once.

Now, the spinner could land on either win or lose.

And you can see there we've got our set of possible outcomes.

So, inside the curly brackets, we have either winning or we have losing.

Referring to all those outcomes, we refer to that as our sample space.

Ah, hello Jun, what are you gonna be telling us about? Is it possible to draw with this spinner? No, it's not one of the possible outcomes.

So, draw is not included in sample space because you either win or you lose, there's no draw.

Andeep's gonna spin this spinner once.

What are the possible outcomes? So, remember the trial is we spin the spinner once.

What do you think the outcomes could be? So, if you correctly said the spinner could land on A, B, or C, well done.

Even better, if you could have written the set of possible outcomes using the correct mathematical notation.

So, you can see there we've got C is equal to A, B, and we've got C as well.

So, "Is it possible to get any of the letters of my name?" says Jun.

What do you think? Can we have a J, a U, or an N? Exactly, Andeep, we can't.

None of the letters from Jun's name are in our sample space, and that's because they're not one of the possible outcomes.

Izzy's gonna spin this spinner once.

What are the possible outcomes? Well done if you wrote that the set of possible outcomes could be 1, 2, 3, or 4.

Ah, Jun's back again.

So, Jun's asking, "Is it possible to get a cube number on this spinner?" So, you have to think about your understanding of cube numbers.

Can we get a cube number on this spinner? Exactly right, Izzy.

1 is a cube number and 1 is in our set of possible outcomes, so yes, getting a cube number is indeed a possible outcome.

Is it possible to get a negative number? No, 'cause there are no negative numbers in the sample space.

They are not possible outcomes.

1, 2, 3 and 4 are all positive numbers.

So, Alex and Sophia are considering the sample space for a trial consisting of one spin of this spinner.

Alex thinks there are four possible outcomes.

Sophia says, "I think there are two possible outcomes." Who do you agree with? Well, the only possible outcomes on this spinner are win and lose.

There are multiple sectors on the spinner with each outcome.

So, we can see four sectors, but the sample space only contains two distinct outcomes.

You can either win, which appears on two of the sectors, or you can lose, which appears on the other two.

Sample spaces could also show the outcomes of trials involving other objects, so it doesn't just have to be spinners.

So, for example, a trial could be flipping this coin once, and the set of possible outcomes are heads and tails.

Another trial could be rolling a regular six-sided dice once.

And our set of possible outcomes will be the numbers 1 to 6.

The contents of a sample space can depend on what is defined as a trial in a particular context because this will affect the possible outcomes.

So, for example, Aisha, Lucas, and Sam are going to play a board game, and you might recognise this board game.

So, a trial is defined here as a roll of the dice.

The possible outcomes are the numbers on the dice.

So, our sample space would be the numbers 1, 2, 3, 4, 5, and 6.

If a trial is defined as a game, then the outcomes are the possible winners.

So, in this case, our sample space would be either Aisha, Lucas, or Sam.

They're our possible outcomes because they're the people that are playing the game and so they're the ones that might win.

It's time for a quick check.

The sample space shows the outcomes for a single spin of the spinner.

Fill in the blank for the sample space.

Pause and do this now.

Welcome back.

What letter did you put there in the gap? You should have put S because now our sample space has the outcomes I, N, S and P, which matches our spinner.

The sample space here shows the outcomes for a single spin of this spinner.

Fill in the blank on the spinner.

So, what should go in that final sector? Pause and do this now.

Welcome back.

Did you spot what one was missing? That's right, ears was.

It was the only one in our sample space that wasn't on our diagram.

A trial here consists of one spin of this spinner.

Write down the sample space for this.

Welcome back.

What did you put in your sample space? You should have put back, forward, left, and right.

It doesn't matter what order you wrote these in, so it's absolutely fine if you have these four but in a different order, it's still correct.

It's now time for your first task.

In question 1, a trial consists of a single spin of the spinner you can see.

Write down the sample space for each trial.

So, there'll be different sample spaces for A, B, C, and D.

And then in question 2, a trial consists of a single spin of the spinner.

Use the sample spaces to complete each spinner.

Now, there may be more than one possible answer, so please don't worry if your answer doesn't agree with someone else, you might still be right.

Pause and do this now.

Welcome back.

Let's go through our answers.

So, for 1a, the sample space should be 1, 2, 3, 4.

For b, it should be A, B, C.

For c, it should be win and lose.

And then for d, did d catch you out? It's just win and lose, it's exactly the same as c.

Remember, although we have more sectors showing win and more sectors showing lose than we do in c, there's still only two possible outcomes.

In question 2, you had to complete the spinner.

Note, the letters could go in any order here, so you don't have to have A in the same place I have A, but you do need each sector to contain a unique letter and it should go A, B, C, and D.

Now, in b's case, we've got A, B, C, D have to go on the spinner, but you can repeat whichever letter you like.

You do need all four to be there at least once, but it doesn't matter which ones you repeated.

Time for the second part of our lesson, and in this part we're gonna be recognising when two outcomes are equally likely.

Hello, hi Jacob.

He's gonna spin one of these spinners and if the spinner lands on win, he gets a prize.

Nice, okay.

Oh, Jacob's a little concerned about these spinners.

He says that one of them doesn't seem fair.

Which spinner seems unfair? And why do you think that? Now, likelihood describes the chance that particular event will occur.

For the spinner on the left, the sectors for win and lose are the same size, so the likelihood that the arrow lands on the win sector is the same as the lose sector, so they're equally likely to happen.

In the spinner on the right, the sector for losing is larger than the sector for winning.

So, in terms of is it equally likely that I'll win or lose? The answer is no.

Alex and Sophia are trying to decide who should get the last cookie in a jar.

They decide to roll a regular six-sided dice to determine who gets it.

So, remember our possible outcomes here are Alex gets the cookie or Sophia gets the cookie.

Alex says, "If we roll a multiple of 3, then you can have the cookie.

If it's not a multiple of 3, then I'll have the cookie." Hmm, is this fair? So, in other words, what we mean by this is are the outcomes equally likely to happen? So, in other words, is there an equal chance of Alex getting the cookie as there is Sophia? No, there's not.

Why is that the case? Well, Alex gets the cookie if the dice shows a 1, a 2, a 4, or a 5, whereas Sophia only gets the cookie if it shows a 3 or a 6.

In other words, there are four outcomes that work for Alex and only two that work for Sophia.

Lucas and Jun are trying to decide who should get the last cookie in this jar.

They decide to roll a regular six-sided dice too.

Okay, Lucas, you reckon if we roll an even number then Jun gets the cookie.

And if it's an odd number, Lucas gets it.

Do you think this means they both have the same likelihood of getting the cookie? Yes it does because for Lucas he can roll a 1, a 3, or a 5 and he gets the cookie, and Jun gets the cookie if it's a 2, 4, or a 6.

There are for each of them three outcomes that allow them to get a cookie.

Aisha's gonna flip a fair coin and note whether she gets heads or tails.

She's already flipped the coin nine times and here's what she got.

So, you can see her results there.

She's about to flip the coin again and she's thinking about what she expects to come up, "It's been a while since tails came up last, so I reckon the next flip will be tails." What's wrong with Aisha's reasoning? Now, there are two possible outcomes and they're equally likely.

It doesn't matter what came before, it's still equally likely that she'll get heads or tails.

So, let's do a quick check.

For which spinner are the outcomes win and lose equally likely to happen, spinner a, b, or c? Pause and pick the one that you think has equally likely outcomes.

Welcome back.

Did you pick c? If you did, well done because in C, we have three sectors that are lose, three sectors that win, and all sectors are the same size.

This tells me that c is the one where our outcomes are equally likely.

Time for our next task.

I'd like you please for question 1 to tick the spinners where the outcomes win and lose are equally likely.

Pause and do this now.

Welcome back.

Let's go through our answers.

So, we should have ticked spinner number 3 as I go across because there I have three sectors for win, three for lose, and all sectors are the same size, so the outcomes are equally likely.

The next spinner will be in the second row and it's the second one along, and that's because half is for winning and half is for losing.

You can see that with our circle, so although there are two sectors for lose, they take up the same amount of space as the sector for winning does.

And then, the very last spinner also had equally likely outcomes.

And this is because I had three sectors for win, three for lose, and they're the same size.

The spin again didn't affect it, win and lose was still equally likely.

Time for the final part of today's lesson, and in this section we're gonna be recognising more than two equally likely outcomes.

A charity raffle gives each person a number from 1 to 100.

A number is then randomly picked from a box and the person with that number wins a prize.

Cool.

Jun says, "I've got number 53." Oh, Laura does not seem happy.

She's got the number 1, "That's never gonna get picked! Jun's far more likely to win than me," says Laura, "because he has a number in the middle." Hmm, not sure how I feel about that, Laura.

I think she's wrong, do you agree? Why might Laura be wrong? Laura's definitely wrong and it's because each number is equally likely to be picked regardless of its size.

So, picking the number 1 is just as likely as picking the number 53.

Andeep is gonna spin this spinner once, are all the outcomes equally likely to happen? So, you can see the set of outcomes there and you can see the spinner.

What do you think? Are all the outcomes equally likely? Andeep says, "All the sectors are the same size and there is one sector for each letter.

So, yes, the outcomes are equally likely to happen," and Andeep is spot on with that.

What about this spinner? Are all the outcomes equally likely here? What do you think? Pause the video while you discuss this.

Andeep says, "All the sectors are the same size and there are two sectors for each letter.

So, yes, the outcomes are still equally likely to happen." And Andeep's spot on there.

What about this spinner? Pause and discuss.

Right, what did you say? Let's see what Andeep says, "All the sectors are the same size.

Ah, but A and B have three sectors each, but C has less than that, it only has two.

So, no, the outcomes are not equally likely to happen." A and B are equally likely to happen, but not A, B and C, so it's not all of them.

What about with this spinner? Andeep says, "Well, there are more sectors with B than the other letters," and he's right, "but the sectors for B are smaller.

Each large sector is the same size as two small sectors.

So, yes, our outcomes are equally likely," and Andeep is spot on there.

So, in other words, two sectors that show B are equal to one of the sectors that shows either A, D, or C.

So, yes, we have equally likely outcomes yet again.

Right, time for a quick check.

The different outcomes for this spinner are equally likely to happen.

So, have a look at the spinner.

Do you think that's true or false? And then, you need to pick either statement a or b to justify the answer you've chosen.

So, either justify your answer with the sectors are all the same size and there's the same number of sectors for each letter, or justify your answer by saying some letters are more difficult to get on the spinner than others.

Pause and make your choice now.

Welcome back.

You should have said that this is true.

The different outcomes for this spinner are equally likely to happen, and you should have justified it by saying statement a.

The sectors are all the same size and there is the same number of sectors for each letter.

What about this spinner? The different outcomes for this spinner are equally likely to happen.

Is that true or is it false? Now, note the two statements that justify, one of them's changed, so you do need to read these carefully.

A now says even though one sector is bigger than the others, there is the same amount of space on the spinner for each letter.

And b says, some letters are more difficult to get on the spinner than others.

Pause and make your choice now.

Welcome back.

You should have said that it's true and that justification is a.

So, even though sector A is bigger than the others, the same amount of space on the spinner for each letter is true.

So, in other words, adding the two sectors that are C is equal to the sector that's A, and the same is true for B.

Time for our final task now.

Question 1 says complete the spinners to make the outcomes equally likely to happen.

So, you need to fill in your diagrams very carefully, making sure that of the outcomes, they're all equally likely.

Pause and do this now.

Welcome back.

Question 2, Aisha, Jun, and Izzy are playing a game that involves rolling a regular six-sided die.

Aisha's piece moves if the dice roll is a 1 or a 2, Jun's piece moves if the dice roll is 3 or 4, and Izzy's moves if the dice roll is 5 or 6.

Izzy thinks that 6 is the most difficult number on the dice to roll.

Explain why this is incorrect.

And then, in part b, the sample space shows all the possible outcomes for each trial, so either Aisha moves or Jun moves or Izzy moves.

Explain why these outcomes are equally likely to happen.

Pause and do this now.

Welcome back.

Let's go through our answers.

So, in question 1, you had to complete the spinners to make the outcomes equally likely.

So, for question 1a, you needed two sectors that showed A and two that showed B.

They don't have to be in the same places that mine are.

In b, we needed three that showed win and three that showed lose.

And again, it can be in different places to mine.

And then, for c, we needed any four even numbers and any four odd numbers, and you could have repeated those numbers.

So, you could have said, "I'll have four segments that have the number 2 in and four that have the number 1 in," that would've worked too.

Indeed, you needed to show green and blue, and that means that one of the sectors had to show one colour and the two smaller sectors had to show the other, so you may have switched that round.

For e, we needed to have vowels and consonants and they needed to be equally likely.

Don't forget, of course, you could have repeated a letter.

The large sector is the same size as two small sectors, so what was important was we needed to make sure that the large sector and one of the small sectors had either both got vowels in or both got constants in.

And then the other three smaller sectors had to be the opposite.

So, if you'd already used vowels, you need to use consonants on the other side.

And then in f, this one was a little bit harder, the largest sector is the same size as four of the smaller sectors.

And then, the next largest is the same size as three smaller sectors.

So, what we needed here was one letter to go on four smaller sectors and then a different letter for a small, and the one that looks like a quarter of the circle, and then the largest sector had to be the last remaining letter.

Right, question 2.

So, Izzy thinks that 6 is the most difficult number to roll.

She's incorrect because each number on the dice is equally likely to be rolled.

And then in b, we had our sample space and you had to explain why these outcomes were equally likely.

Now, each player has the same number of outcomes which moved their piece.

And each outcome, that's the number I roll on the dice, is equally likely to be rolled.

It's time to sum up what we've been learning today.

So, we've used quite a few technical words today, and it's worth just going over what these mean.

So, a trial is a single predefined test.

An outcome is a result of a trial.

A set of possible outcomes for a trial can be stated by writing a sample space.

Outcomes can be evaluated to decide how likely they are to happen.

And by considering the likelihood of outcomes, we can identify when outcomes are equally likely to happen.

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

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