Lesson video

Hello, I'm Mr. Langton.

And today we're going to look in more depth at combined events and sample spaces.

All you're going to need is something to write with and something to write on.

Try and make sure you're in a quiet space with no distractions.

And when you're ready, we'll begin.

We'll start with the try this activity.

Cala and Zaki are playing a game where a six-sided die is rolled at the same time as spinning the spinner.

Who is more likely to win? And how many wins would you expect from each person in a hundred games? Cala says that she's going to win if the spinner lands on purple and they roll a multiple of three.

Zaki's going to win if the spinner lands on white and they roll a factor of 24.

Pause the video and have a go.

When you're ready, unpause and we can go through it together.

You can pause in three, two, one, go.

How did you get on? There was a lot to consider with that question.

For a start, Cala can win with a purple spinner of which there were two options and a multiple of three, of which there were also two options.

Zaki could win if you got a white, of which there was only one option and a factor of 24.

Well, there were five factors of 24 that were there.

So we need to work out who is most likely to win.

Is Zaki most likely to win because he's got six possible ways to win? Does he have about five ways? Has Cala only got four ways? We need to visualise it a little bit better.

And the way that we're going to do that is to draw a sample space diagram.

We'll start with the dice.

It can come with a one, a two, three, four, five, or six.

And a spinner, I'm going to put it down on the side.

There were two purple options, two green options, and a white option.

What I'm going to do is fill in all the possible outcomes that we could have.

So we could have a one and a purple, a two and a purple, a three and a purple, and so on.

So Cala wins if it lands on purple and she gets a multiple of three.

So that means this one and this one, and this one, and this one.

Cala does have four ways to win.

And we can see that all together, there are 30 possible options that could happen.

Now, Zaki comes next.

He wins if it lands on white and it's a multiple of, sorry, a factor of 24.

Factors of 24 are one, two, three, four, and six.

So that's one way, two ways, three, four, there are five ways that Zaki can win.

And again, that probability is out of 30.

You could simplify those fractions, but it's much easier to keep them both out of 30 because we can compare.

We can see that Cala has got a four out of 30 chance and Zaki has a five out of 30 chance.

So keeping the fractions with the same denominator actually makes it much easier in this case.

Okay, now it's your turn to have a go.

Pause the video and access the worksheet.

Have a go at the questions and see how much you can do.

And when you're finished, we'll go through it together.

Good luck.

How did you get on? I completed the sample spaces for you here.

Now let's go through the questions.

Question one, part B.

What is the probability I get at least one three? So all the ways that I can get a three.

I want to circle these now.

That's one, two, three, four, five, six.

There are seven ways I can get a three.

And all together, there are 16 options, so it's seven out of 16.

Next up.

What's the probability that both of my numbers are even? So let's have a different colour, let's go for green.

So both numbers are even.

So I could have a two and a two, a four and a two, a two and a four, and a four and a four.

So if both my numbers are even, there are only four ways that can happen.

And again, that's out of 16.

I could cancel it out if I wanted to, to a quarter, but if I wanted to compare which is most likely, I want to keep them both in 16ths because seven 16ths can be compared to four 16ths much easily, easier.

What is the probability that I get a three or a four on at least one of the dice? So first, I'm going to need to make a bit of space.

I've scribbled over that too much.

The probability that I get a three or a four.

So, if I get a three on at least one dice.

So that's all those possible options there.

That's the first dice, I get a three or four.

And that's all of these here for the second dice.

So at least one three and one four, I've coloured in 12 of the options all together out of 16.

Or three quarters.

Okay, question two.

The probability that I get at least one black.

So at least one black.

So all of these have at least one black and all of these have at least one black.

So all together, that's 10, 11, 12, 13, 14, 15, that's 16 that I've coloured in or I've circled out of 25.

The probability that I get a black and a white.

Well, let's get rid of what I've done there so we can see just a little bit easier.

Exactly one black and one white.

So that's going to be these six options and these six options.

So there are 12 options, out of 25.

What is the probability that the spinners are going to land on either both black or both white? Well, that's going to be the inverse, isn't it? That's our complimentary probability there.

So we could say that it's going to be one take away 12 out of 25, which is going to give me 13 out of 25.

Because if we had just worked out the probability that we get a black and a white, then both black or both white is the opposite of that.

It's everything that's leftover.

We can check that on the grid.

There are nine ways that we can get both white and four ways that we can get both black.

And all together that makes 13, so we've got it right.

We're going to finish off with the explore activity.

This is quite an open-ended task and there are lots of possible answers that you can have.

I'd like you to pause the video and have a go.

When you're ready, unpause it and I'll show you some of the answers that I got.

What you've got to do is for each of the games, draw a sample space and try and think of as many ways that you could win, as many rules that you could have for winning a game.

And ideally, we're going to try to come up with options where there's a 50/50 chance, there's an even chance of winning or losing, a chance where you're better off winning and losing, and a chance where you're more likely to lose than win.

See if you can do that for each game.

Pause the video now.

When you're ready, we can go through it together.

Pausing in three, two, one.

Okay, I want to show you some of my answers.

So I worked out of my sample space, and in game one, the chance of getting numbers one to six is equal of getting eight, between eight and 24.

So that's one way that I've got an even chance of winning.

I could win with a number between one and six.

That's got an equal chance of any number between eight and 24.

I'm more likely to win, well, there were more even numbers than odd numbers that came up with my answers, so I'm more likely to win if I go for an even number, which means I'm obviously more likely to lose if I go for an odd number.

Now, for the second game, spinning the two spinners.

So again, where I have an even chance of winning, it was a little bit tricky to come up with something.

I said that the probability of scoring a green is equivalent to the probability of scoring a red.

So if we say that you win by choosing red and you lose by choosing green and anything else is a draw, then I've got an even chance of winning there.

I found that one quite tricky.

A better than even chance of winning.

Well, the probability, if I go for a win, it's either red or blue, that has got a greater chance of winning than the other ones with the white and the green.

So if we played that spinner game and I said that I win if I get a red or a blue, and you win if you get a white or a green, I'm more likely to win.

And on the last one, we could flip that around to the other way, couldn't we? So the probability of getting a white or a green is less than the probability of getting a red or a blue.

That's one possible option.

You may have come up with others.

I'll leave that there for now.

I'll see you later, goodbye.