Lesson video

Hello.

I'm Mr. Langton.

Today, we're going to look at combining events and sample spaces.

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

Try and find a quite space where you won't be disturbed.

When you're ready, we'll begin.

We'll start with a try this activity.

Binh and Antoni are playing a game with two six-sided dice.

They roll both dice and add their results.

Who do you think has a better chance of winning? Binh will win the game if the dice add up to six, seven, eight or nine.

Antoni will win if the dice add up to anything else.

Pause the video.

Have a think about it.

See if you can work out who you think is going to win.

When you're ready, un-pause it.

We can go through it together.

You can pause in three, two, one.

When we're combining two events, such as rolling two dice, we can represent the probabilities in a sample space.

So first, we need to fill in all the results that can happen On the top, we've got the first dice.

Down the side, we've got the second dice.

So should the first and the second dice both roll as a one, then we'll score two points.

Should the first one be two and the second one be one, we'll score three points and so on like that.

For example, over here, in this square here, the first dice is five, the second one is three we've got a total of eight.

Now I'm going to cheat and just fill in the rest of them now.

There we go.

So that's all the possible scores that we can get.

So the lowest score we can get would be two and the highest score we get can be 12.

And this doesn't just show us all the scores, it shows us how many times each score can come up.

So, binh says that she'll win if she gets a six, a seven, an eight or a nine.

So we start looking at the sixes.

There's one, two, three, four, five sixes.

She said seven, there's one, two, three, four, five, six sevens.

Eight, one, two, three, four, five chances of getting eight.

And nine can come up one, two, three, four times.

So all together, that's 20 times that binh can win.

I'm just going to rip out that we finish change that colour.

Because every other number that comes up will be one where Antoni can win.

Now we can count them all if we want but since we know that there are 36 outcomes all together, If Binh can win 20 ways, then Antoni can win 16 ways.

So the probability of Antoni winning is 16 out of 36 and the probability of Binh winning is 20 out of 36.

So Binh is most likely to win.

Now it's time for independent task.

There are two questions for you.

The first one, you're going to have to fill in you're own sample space and use it to calculate the probabilities.

The second one, we're going to use the sample space from earlier with two dice and you're going to calculate some probabilities from that.

Good luck.

How did you get on? Let's go through some answers together.

So I'm starting off by tossing two coins.

So I can only get head followed by a head or a head followed by a tail or a tail followed by a head or a tail followed by a tail.

What is the probability of getting two heads? There's only one way that can happen so that is going to be one out of four because there are four ways all together.

Four possible outcomes.

What's the probability of getting a head and a tail? So I could either get a head followed by a tail or a tail followed by a head.

So there are two ways that can happen out of four.

Which is a half.

Now you can see, the only other thing that could happen Lets go for green.

Is tail, tail which is a quarter.

So if we've got a chance of a quarter for tail, tail, we've got a probability of a half for a head and a tail and we've got the probability of a quarter for two heads and all together, that makes a whole one.

It covers all the possible outcomes.

Okay, lets look at the second question.

I roll two dice and add the scores together.

What's the probability of scoring a two? There's only one way that I can score a two.

That's with a one and a one.

And there are 36 possible choices all together.

So it's one out of 36.

The probability of scoring a six.

There are one, two, three, four, five ways that I can score a six out of 36.

More than an eight.

So that's one, two, three, four, five, six, seven, eight, nine, I believe there are 10 ways you can score more than an eight out of 36.

We can simplify that if we want to five out of 18.

An even number.

One, two, thee, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, I think there are 17 even numbers out of 36.

A factor of 12.

So, factors of 12 are one, two, three, four, six, and 12.

Lets count of how many ways they can all come up.

So we can't score a one at all.

There's one way we can score a two.

There are two ways we can score a three.

There's one, two, three ways that we can score a four.

We've already said there are five ways you can score a six.

And only one way you can score a 12.

So one, two, three, six, 11.

There are 12 ways that we can score a factor of 12 out of 36.

Which simplifies down to a third.

So, finally, we've got the explore activity.

Binh and Antony are playing a different game with two six-sided dice.

They roll both dice and work out the difference between their results.

Which are the best numbers to pick to win? And why? What is the probability of winning on any roll if you have the best three numbers? Pause the video and have a go.

When you're ready, un-pause it.

We can go through it together.

You can pause in three, two, one.

To help me get the answer, I have to run a sample space showing the difference in the scores between each dice.

For example, if I roll a three on my first dice and a one on the second dice, the difference between those numbers is two.

So what I'm going to do is see how likely each result is to come up.

I can score.

I have to draw a table.

I can score either zero, one, two, three, four, or five.

Now, how many outcomes are there for each one? So I can score a zero in six ways.

I can score a one in 10 ways.

I can score a two in eight ways.

I can score a three in six ways.

I can score a four in four different ways and I can score a five in two different ways.

The first thing I'm going to do is just check that adds up to make 36 just so I'm not making a mistake Six and 10 is 16.

16 and 8 is 24.

Add six is 30.

34, 36.

Good.

So we definitely got all the possible outcomes.

Now, the question said, which are the best numbers to pick? So, one is clearly the best number to pick.

It can come up 10 ways.

So the probability of getting a one is 10 out of 36.

The next best way would be two.

With a probability of eight out of 36.

And then a zero and three have both got the same chance of happening.

Six out of 36.

Four is four out of 36.

Which is a ninth.

And five is two out of 36.

So what is the probability of winning on any roll if you have the best three numbers? That's a little bit misleading because you could argue what are the best three numbers.

Clearly the best number to have is one.

The next best number to have is two.

The next best number is either zero or three.

They've both got the same chance at winning.

In that case, the probabilities of me winning, we're going to have 10 over 36, creating a one.

We're going to have eight over 36 for a two.

And we're going to choose zero or three as your next best option.

It's six out of 36.

So all together, the probability of winning if I have those numbers will be 24 out of 36.

Which is a two thirds chance of winning if I can pick the numbers that are going to win for me.

That's it for today.

Thank you for going along with me.

I hope it all made sense.

I'll see you later.

Goodbye.