This frequency tree is incomplete.

How many boys didn't do their homework?

How many girls were there altogether?

How many students were there altogether?

What is the probability of randomly selecting a student who did their homework?

Given that I select a girl, what is the probability that she didn't do her homework?

Frequency Trees

﻿Hello, I'm Mr Langton. And today we're going to be looking at calculating the probability of combined events. 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. The two samples based diagrams represent outcomes of spinning two spinners together. What's the same, and what's different about the two different diagrams. Pause the video and have a think about it. Maybe make some notes when you're ready. I'll pause it. We'll discuss a few ideas together. You can pause in three, two, one, go. Okay. I'll give you a few ideas that I came up with. Um similarities. They both show all the possible outcomes. They're fair to all the colours that are there, and you can use them to contact the probabilities. They're both useful in their own way. Uh different. Um in the one on the right, each colour is only represented by the fraction of the times it comes up. So on the left, for example, spin a one. We've got blue, green, pink, green, green, pink. So we've got all six individual options, whereas on the right green comes up half of the time. So it takes up half of the space. Pink comes up a third of the time. So it takes up a third of the the space. And blue comes up one six of the times, it takes up one sixth of the space. So we're looking at the area of the shapes and the area of those rectangles is going to represent the probabilities. We'll look at that in a bit more detail in the next slide. Now I'm going to try and use both of these diagrams to answer some probability questions. Parte calculate the probability that both spinners landed on green. Now the diagram on the left, we're looking for every, every time green comes up on both of them. It's not one there, not one there, and not one there. So that's three distinct ways that they can both land on green. And there are 18 possible options, which means that that's one sixth of the time. Now, if you look at our diagram on the right hand side, the only parts of it that is green and green, is this rectangle in the top left. And the area that rectangle is a half multiplied by a third, which makes a sixth, just like we got there. So we can use both diagrams to get the same answer. It's really useful to remember the calculation that we did because later on, we won't get diagrams and we'll need to be able to run with these methods. Part B at least one option is green. So at least one green I'm going through here, and I'm circling all the possible ones. There is at least one green. Now don't forget. That will include both of them being green. So then keep those two blue ones on there, I'm sorry, there's three blue ones on there as well. Three, six, nine, 10, 11, 12, So there are 12 options, where there is at least one green. Now to the 18, and that's two thirds. Now I can do that with the diagram as well. And sometimes you might find it easier to get a common denominator along the bottom. So that's working six. So that would be three-sixth and a half is the same as three-sixth. And a third is the same as two-sixth. We want at least one green. We've already said that that one on the top left is the sixth. The next one. In fact, do you know what I'm going to change that whilst it is a sixth, I'm going to make sure that I keep my common denominator all the way through. So I'm going to do one third multiplied by three-sixth, because that's going to give me three over 18. And it means every single time that I do this, I'm going to get the same denominator. So here, I'm going to do two six multiply by a third, which is two over 18. You can see that. And the last one is a third multiplied by a sixth, which is one over 18. Now also we can go down here. We've got three. So we've got again, one third multiplied by three-sixth, which is three over 18. And again, one third multiplied by three sixth, which is three over 18. If I add all these probabilities together, I've got one 18th, two, three 18th, four, five, six, seven, eight, nine, 10, 11, 12, once again, I've got 12 eighteenths, which is two thirds. Now in this particular example, I think that the first diagram was easier to use. But once again, that method of multiplying the two probabilistic probabilities together is going to be really important later. Finally, the problem is that we get exactly one blue, I'm just going to erase this. So I've got a bit of space to do some more working out, and it's going to take this time to remind you that anytime you do a test, don't you ever erase anything that you've written. You leave everything on that because you can always pick up marks. I've just run out of space and I need to do an extra question. They will always give you enough space on your exam, right, so we're ready now. Problem two. We get exactly one blue. That's going to rule out that option in the bottom left here. That one's no good because it's got two blues. So exactly one blue. And that's going to give us this box here and the one underneath it and the two along the bottom as well. So let's look at this probability from here. First, we're doing one third multiplied by three sixth. Which is going to be three eighteenths. And this one here, we're doing a third multiplied by two sixths. So that's going to be two 18ths. We've got in the very, very top one sixth, multiplied by one third, is one 18th. And again, one 18th. That's going to give me one, two, three, four, five, six, seven 18ths. Let's check that with this other diagram, the one on the left and see if we're right. Exactly. One blue. That's got one blue. That's got one blue. That's got one blue. That's got one blue That's one blue, that's got one blue. That's got one blue, and that's got one blue. I don't think I've missed anything. One, two, three, four, five, six, seven. Yep. Okay. That's good. So both methods get me the same answer. That's going to be really important later on. Now it's your turn type of go pause the video, access the worksheet. And when you're ready, we can go through it together. Good luck. Okay, let's go through the answers. The first question. I spin two spinners that are equally likely to land on a one, a two or three. Complete the probabilities for the sample space. So starting the top left probability to getting a one is a third and probability to getting a one is a third, one third multiplied by one third is one ninth. Now in this case, all the probabilities are the same. It makes it quite an easy one to fill in. And it's also going to make it relatively easy to do the maths afterwards, especially as the denominators are all the same. Calculate the probability that both spinners score greater than one. So it's like two different colours. So any spinner that scores one is no good. So I can start to cross those out. So these are the four that are left. In all of these cases, both spinners score greater than one. The probability that both spinners, I guess, score greater than one, it's going to be four out of nine. Okay. Now for the second question. Complete the probabilities for each outcome and what is the probability that I pick one of each colour and I start off, I need finish labelling it down today. In Bag A, one fifth is one one fifth is white. So let's label that white and label that green. Bag B once again, I've got whites and I've got greens. There are six whites and there are 14 altogether. Let's just simplify that probability to three sevenths. There are eight greens in that second bag, and there are 14 altogether that will cancel down to four sevens. The probability that I choose two whites, there's going to be a fifth multiplied by three sevenths. Which is going to give me three out of 35. And probability that I choose a green in that first bag, is four fifths. Multiply that by the probability that I choose a white in the second bag, I'm getting 12 out of 35, probability that I get a white in that first bag, multiplied by the probability that I get a green in the second bag, is going to be four out of 35. And finally, probability that I get a green in both bags or the first bag. The probability is four fifths. And the second bag, the probability is four sevenths, which gives me 16 out of 35. So I've completed the sample space. What is the probability that I pick one of each colour? So, that means I finally get a white and a green, or I can get a green and a white. Then we need to add those probabilities together. So it's four over 35 and 12 over 35. The probability that I get one in each colour is 16 out of 35. We'll finish up with the Explorer activity, two bags, hold green and white cubes. Cubes are drawn from both bags and then replaced. The probability of drawing a green cube from both bags is 0.18. And the probability of drawing a white cube from both bags is 0.28. What is the probability of getting one green or one white? And how many cubes could be in each bag? This is a really tricky question. If you're feeling super confident, you could pause the video now and have a go. If you want a little bit of help, I'm going to give you a hand. So I'll just say now, if you want to start pause it in three, two, one, and if you want a hint, let's keep going just little bit longer. I'm going to draw a sample space for everything that could happen. So I could get the green or a white from one bag and a green or a white from the other. The probability of getting two greens is 0.18, and the probability of getting two whites, is 0.28? Can you think of some decimals that will multiply together to make these? So what two decimals could multiply together to make 0.18? What numbers could multiply together to make 0.28? And can you find some numbers that are going to work for everything? So there's your hint. Pause it and have a go. Pausing three, two, one. Okay. Let's look at some possible answers then. So to multiply 0.18, we could do 0.1 multiply by 0.18. We could do 0.2, multiplied by 0.9. And then I got the first one wrong, haven't I? That wouldn't work. We could do one multiplied by 0.18. We could do not 0.2 multiplied by not 0.9. We could do not 0.3 multiplied by not 0.6. If we were doing not 0.1, no, then that would have to be multiplied by 1.8. And that's impossible because we can't have a probability greater than one. So I'm pretty confident that I'm looking at one of these pairs here. So let's try some out. Let's say that that's 0.3 and that's 0.6. So looking along the top, if the probability of getting a green is 0.3. The probability of getting a white must be 0.7. Looking down the side. If the probability getting green is 0.6, the probability of getting white, is 0.4. Now let's see, let's multiply those whites together. 0.7 times 0.4. Is 0.28. So I've got my probabilities correct. I can now look at what the other probability is going to be. That's going to be not 0.7 times not 0.6 is not 0.42, and not 0.4 times not 0.3 will be not 0.12. Now, how many cubes could be in each bag? Well, in this bag here, looking at this side. The probability is the 0.6 and 0.4. So that's equivalent to three fifths and two-fifths. Which means however many cubes are in that bag, it must be a multiple of five. It must be in the five times table. So the fraction councils down to three fifths. Looking at the other bag, the one that we've got going along the top, we've got two fractions in there. We've got three tenths and seven tenths. So this bag, the number of cubes inside must be a multiple of 10. That's it for today. Thank you for following along and thank you for having a go. I'll see you later. Goodbye.

I roll a fair six-sided dice. What is the probability I roll a 4 or a 6?

I spin two three sided spinners.

What is the probability that both land on 2?

What is the probability that they land on the same number?

What is the probability that they land on different numbers?

What is the probability that their sum is greater than 3?

Calculating the probability of combined events

In this lesson, we will look at how we can use area maps to calculate probabilities of combined events.

Maths

Y9 Maths

Key Stage 3

Year 9

Probability

Statistics and probability

THEME

UNIT

Representing probability

Complementary events

Calculating probabilities

Theoretical probability and relative frequency

Theoretical probability and relative frequency (Part 2)

Comparing probabilities

Combined events and sample spaces (Part 1)

Combined events and sample spaces (Part 2)

Frequency trees

Combined Events and Tree Diagrams (Part 1)

Combined events and tree diagrams (Part 2)

Cala rolls a biased dice 40 times. It comes up with a three 14 times. Calculate the relative frequency of rolling a three.

Binh throws a biased dice 200 times. The theoretical probability of it landing on a 2 is 0.6. How many times would you expect her to throw a 2?

Yasmin has 200 daisy seeds to plant in her garden. The probability of a daisy growing is 0.8. How many daisies would you expect to grow?

There are 18 marbles in a bag. 13 of them are not green. What is the probability of selecting a green marble?

There are 100 cars in a car park. 40 of them are yellow. What is the probability of choosing a car that is not yellow?

In a group of ten people, the probability that at least two were born on the same day of the week is 1.

What is the probability that this fair spinner lands on an odd number?

What is the probability that this fair spinner lands on a white triangle?

I roll a dice 50 times and it lands on a "six" 38 times. What is the relative frequency of getting a "six"?

True or false: I am more likely to flip a coin and get "Heads" than I am to roll a six-sided dice and get a 4.

True or false: I am more likely to roll an even number than a prime number on an eight sided dice.

Lesson overview:Calculating the probability of combined events

Core Content

Probability: