Lesson video

Hello, I'm Mr. Langton and today we're going to look at combined events and probability trees.

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.

Two bags contain green and white cubes only.

The sample space shows the probabilities of getting different combinations of colours, if one cube is drawn from each bag and then replaced.

What probabilities can you state using the sample space? What can you say about the number of green and white cubes in each bag? What I'd like you to do now is pause the video and have a go.

When you're ready unpause it, and we can look at the answers together.

You can pause in three, two, one.

Okay, what did you come up with? Looking at bag A first, the probability of the greens coming up is 5/6, so that means it must be a 1/6 probability that I'll get a white.

And it's also going to tell me, that there must be, at least six cubes in bag A.

But actually, bag A must be a multiple of six.

So it could have six cubes, it could have 12 cubes, it could have 18, it could have 24.

And the reason for that is that when I cancelled out my fractions, they simplified out to six.

So at the very least there must be six, but it must be something that's in the six timetable.

Otherwise it wouldn't cancel down to six.

Bag B, the probability of getting a green is 2/5, so the probability of getting a white must be 3/5 which means that in the case of bag B, the number of cubes inside must be a multiple of five.

So five, 10, 15, or 20.

Okay, let's move on.

So what we'll do now is look at converting our sample space into a tree diagram.

We're going to start off by focusing on bag A.

When I reach in to bag A, I can draw out a green cube or a white cube.

I'm then going to reach into bag B, where I could draw a green cube or a white cube.

Green cube or a white cube.

That's all the possible things that can happen.

Starting with bag A, the chances of me pulling out a green cube is 5/6.

And the chance of getting a white cube 1/6.

Now with bag B, the probability of getting green is 2/5, which means probably of getting white is 3/5.

Now it doesn't change depending on what happened in the first bag so the probably of getting green is still 2/5.

And the probably getting white is still 3/5.

Now, when we look at the diagram on the left, the probably to getting two greens, we've already said, it's the area of that rectangle isn't it? It's the area of this rectangle here.

And we do that, we get the area that in 2/5 multiplied by 5/6 giving us 10 over 30, which is 1/3.

Now it's the same with the tree diagram, the probability of getting two greens will be about 5/6 going up here, multiplied by 2/5.

So the probability of getting two greens, which I'm going to write like this, the probability of getting a green from bag green, is going to be 5/6, multiplied by 2/5, which is 10/30, or 1/3.

Next up the probability of getting a green followed by a white, is going to be the probability of green in that first bag, 5/6, multiplied by the probability of a white in that second one, 3/5, which gives me 15/30.

And the probability of getting a white followed by a green, probability of white followed by green is going to be 1/6 multiplied by 2/5, which will be 2/30.

And the probability of this last one, of getting two whites, is going to be 1/6 multiplied by 3/5.

Which gives me 3/30.

I just need to check, these are the four possible outcomes I can have.

I just got a blue bubble around these there.

Okay I get two greens, the green and white or white and a green or two whites.

If that's all the possible outcomes then all the probabilities should add up to what number, what number, what number? They should add up to one, shouldn't they? So let's check that.

10/30 plus 15/30 is 25/30, add another two is 27/30 and add that three there gives me 30/30, which is a whole one.

Now we can use this tree diagram in just the same way with the sample space.

If I wanted to see what is the probability that one of them is green and one of them is white, well on the sample space over here, I'd find the area of these two parts and add them together and that's exactly the same here.

The green and white is 15/30 and a white and a green is 2/30, so I would add those two probabilities together.

and it would be 17/30.

Okay, now it's your turn.

Pause the video and access the worksheet.

There are three questions for you.

On two of them I've drawn the three diagrams and you've just got some bits to fill in.

Then you have to do the working out and answer the questions.

The third one you have to draw the probability tree yourself.

Pause the video and have a go.

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

Good luck.

Okay, let's look at the answers.

I flip the coin twice, complete the probability tree.

Now, in each case, when I flip the coin, the chance of getting a head or a tail is a half.

So each individual attempt is a half.

Now every time I complete a tree diagram, I like to consider all the possibilities.

So in this case, that's the probability that I get two heads.

And I find that by doing 1/2 multiplied 1/2 which is 1/4.

And the second one, that's the probability of that I get a heads followed by tales, which once again is 1/2 multiplied by 1/2, which is 1/4.

And I know that you can see a pattern, and you're going to wonder why I'm doing this because you're going to think that every answer's going to be a quarter, aren't you? But it's going to help us in a minute when we get to the next bit.

So the probability of getting two tales, is 1/2 multiplied by 1/2, which is 1/4.

Now let's move on to the questions.

What is the probability of getting two heads? Well I've worked that out.

The probability of getting two heads is 1/4.

What's the probability of getting two tails? We've done that working out and it's a 1/4 What's the probability of getting two different results? So in this case, that could be this one or this one.

So it's two different results.

That's going to be 1/4, add 1/4, which is 1/2 So by doing all the legwork early on, I'm doing all that working out while it was fresh in my head, once it came to answering the questions, A, B and C it was really easy to get through it straight away.

Next up, Yasmin and Zaki each take a test.

The probability that Yasmin passes is 0.

7.

The probability that Zaki passes is 0.

6 So let's fill in the tree diagram.

So the probability that Yasmin passes is 0.

7 so the probability she fails must be 0.

3.

If Zaki's passing is 0.

6, probability he fails is 0.

4.

And that is going to be the same there.

Now, like I said, I like to try and work out these probabilities as soon as I can.

So the probability that we get a pass followed by a pass.

That's going to be 0.

7, multiplied by 0.

6, which is 0.

42.

Probability that Yasmin passes but Zacki fails, is going to be 0.

7 multiplied by 0.

4, which is 0.

28.

So right now I'm not even thinking so much about the question.

I'm just making sure that I've got my working out sorted.

Probability that Yasmin fails and Zacki passes, that's 0.

3 multiplied by 0.

6, which is 0.

18.

And the probability that they both fail will be 0.

3 multiplied by 0.

4, which is 0.

12.

Well, that's great because that was the first question.

What's the probability that they both fail? 0.

12, I just worked that out.

What's the probability that only one of them will pass? So we're looking at, either Yasmin passes or Zacki passes.

So we're going to need to add those two probabilities together.

0.

28 and 0.

18, which is 0.

46 Carla has seven cubes in a bag.

Two of them are green and five are white.

She takes a cube at random, notes down the colour and puts it back.

She then takes a second cube, notes down the colour and puts it back.

So we have to draw a probability tree that will show all the outcomes.

So starting off here.

She takes a cube and it's either green or it's white The probability that it's green is 2/7.

Probability that it's white is 5/7.

She's put the cube back and then she has another go.

And once again it can either be green or white and these probabilities have not changed.

So probability of white is 5/7 isn't it? Right, So if you don't like doing that working out, like I did, at the very least, I recommend you write down what the outcomes are.

So in this case, I'm going to propose she gets two greens, in this case, green followed by white, in this case, white followed by green.

And in this case, white followed by white.

So part B, what is the probability she takes one cube of each colour? So one cube of each colour is going to be these.

These are the two bits that I need to work out.

So we're going to do 2/7 multiplied by 5/7, which is 10/49.

And we're going to do 5/7 multiplied by 2/7, which once again is 10/49.

I need to add those probabilities together to get 20/49.

We're going to finish with the Explore activity.

Complete the probability tree, so that if a student is picked at random, the probability of them having done their homework is 0.

6 and the probability that they failed their test is 0.

4 and the probability that have not done their homework and failed their test is 0.

28.

It's quite a tricky question so take your time.

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

You can pause in three, two, one.

How did you get on? Let's go through it together now.

The first thing that I'm going to do is label the four possible outcomes.

This one here is the probability that somebody did their homework.

So I'm going to say, yes, they did their homework and they passed.

This is the probability that yes, somebody did their homework, but they failed.

This is the probability that no, they didn't do their homework, but they still passed.

And this one is the probability that no, they didn't do that homework and they failed.

That's going to be really important.

It's going to help you me lot when I'm filling in these options.

The probability that someone did their homework is 0.

6.

That means the probability that they didn't do their homework is 0.

4.

Done that one, I can tick it off.

Now the probability that somebody failed their test is 0.

4 That's going to be a combination of this one here and this one here.

We need to add those two together.

So at the moment I can't answer that.

The next bit, the probability that someone has not done their homework and they failed that test is 0.

28.

Well let's start there.

And that now means that this one here must be 0.

12 because these two must add up to make 0.

4.

And now I can start to work backwards to fill in the gaps.

I'm actually going to find it a bit easier to turn some of these into fractions, I think.

So this would have been 6/10.

This would have been 4/10.

This one over here is 28/100.

And this one here is 12/100.

So let's go right down to the bottom.

The probability that somebody failed, having not done their homework.

So that's going to be 4/10.

Let's write this down.

4/10 multiplied by something equals 28/100.

Well, I can see that it's going to have to be 7/10, because 4 times 7 is 28 and 10 times 10 is 100.

So that there as a decimal would be 0.

7 or 7/10.

Now because these two probabilities here have to add up to a whole one, that must be 0.

3, which makes this probability here 0.

4 times 0.

3, is 0.

12.

Now if that's 0.

12 there, then this one here, where are we looking? Let's go for this one.

So we've got 6/10 multiplied by something, is 12/100.

It's going do this one first answer.

So once again, that's going to have to be tenths and that'll be 2/10.

That's going to go there 0.

2, which means that that one there, must be 0.

8.

And the probability that someone did the homework and passed the test is going to be 0.

6 times 0.

8, which is going to be 0.

48.

And we can check if we add together these four probabilities here, we get a whole one.

So we definitely got it right.