I rolled two spinners and completed the sample space diagram below.

What is the probability I spin 2 blacks?

What is the probability that I spin 2 whites or 2 blacks?

What is the probability that I get exactly 1 black?

What is the probability that I get 1 white and 1 red?

What is the probability that I get 2 whites or 2 reds?

Combined Events and Sample Spaces 2

﻿Hallo, I'm Mr.Langton and today we're going to be looking at frequency trees, all we're going to need is something to write with and something to write on. Try finding a quiet space where you won't be disturbed, when you're ready, we'll begin. We'll start with a try this activity. There are 32 students in a class. 17 of these students did their maths homework, and of these 17 students, 15 passed their maths test that week. Of the students that didn't do their homework, only 3 of them passed. How could you represent this information? Could you make it more visual? Could you make it easier to follow? If you chose a student at random, what is the probability that they didn't do their homework and they failed their test? Pause the video and see what you can do. When you're ready, unpause it, we can go through it together. You can pause it in three, two, one. How did you get on? I'm going to show you how to put the information into a frequency tree. We're going to go through the question one line at a time. The first line says there are 32 students in a class, we're going to make a note to the number 32 there. The next line says that 17 of these students, did their maths homework. I'm going to draw two branches coming out of there, showing those that did, the 17 and those that didn't. Now we know there are 32 students altogether, so 15 students didn't do their homework. Of the 17 students that did their homework, 15 passed their Maths test that week, I'm just going to take that 17 and split it up into two parts, 15 of them passed, that means that two of them failed because those two have got to add up to make 17 eminently. Of the students that didn't do their homework, only three of them passed. So we've got two new branches, three of them passed. Now we've got 15 students who didn't do their homework, that means that 12 of them must have failed. So we've now got a nice and visual method, that shows us all the possible outcomes, we've got no missing numbers, which is going to make the question much more straight forward. If you chose a student at random, what is the probability that they didn't do their homework and they failed their test? Well the students who didn't do their homework, that's 15 of them, 12 of them failed so all together, there were 12 students who didn't do their homework and failed. And in total, we have 32 students. So if I pick students at random from the 32, the chance of me picking someone who didn't do their homework and failed their test is 12 out of 32. Or you could simplify that to down to 6 out of 16 or down to three eighths. Now it's your turn to have a go, you can pause the video and to the worksheet, there are two questions I want you to get on with, one of them, I've drawn the frequency tree, you just have to fill it in for the second one, you'll have to draw it yourself. Pause the video and have a go, when you're ready, we can go through it together. Good luck! How did you get on? Let's go through the questions together. We'll start with the first one. A 100 students had some homework. So we're going to label that straight away, that we've got 100 students. I'm going to take that, tell me that you've already done it. 42 of these students are boys, 42 students are boys. Now before I go any further, 100 subtract 42, equals 58. So add on the 58 girls. We're done with that information I'm going to give that a tick, we don't need to pay attention to that anymore. 8 of the 100 students did not do their homework, now we've got a problem here because that says eight of the 100 students so that means that this one here and this one here adds up to make 8. I'm just going to leave that for now, I'm not going to tick here, there's nothing I can fill in coz I don't know whether that's four and four, or if it's five and three, if it's six and two, let's move on and see what else we can work out. 53 of the girls did do their homework so the girls who did do their homework that's 53 girls. So that means if there are 58 girls all together, that means five girls didn't do their homework. I can tick that coz I've done it and now I can come back to this one. If I know all together, eight students didn't do their homework, if five girls didn't do their homework, then three boys didn't do their homework, and that means 42 take away three is 39. 39 boys did do their homework. Now for the second question, we have got to come up with our own frequency tree. 50 students went to visit either Bath or Manchester University. We start off, we have 50 students and they either went to Bath, or they went to Manchester. We're going to put two bubbles there, I can tick that line just to let me know that I'm done. 12 girls and 11 boys went to Bath so, 11 boys and 12 girls. Now that makes in total 23. So that tells me that 27 people went to Manchester so I've got awful lot of information just from that statement. 13 girls went to Manchester. So that means 14 boys went to Manchester. So I've drawn our frequency tree. Calculate the probability that the students picked at random went to Manchester. We know that 27 students went to Manchester, and there were 50 students altogether. The probability that the student picked at random was a girl, we're aware that there were 13 girls in Manchester and 12 girls in Bath so in total that means it's going to be 25 girls. And as there were 50 students, there's also 25 boys. Part D, given that they were a boy, that they went to Bath? That's a really tricky question to try and answer. Altogether, we've got 25 boys so we're looking at the 25 boys, and of those 25 boys, 11 went to Bath and 14 went to Manchester. So that's a really sneaky question to try and answer. How did you get on? We're going to finish with this explore activity. You are to complete the frequency tree, matching these criteria; 3 students passed despite not doing their homework, half of the group did their homework, 8 of those that did their homework failed the test, and twice that number passed the test altogether. Pause the video and have a go, when you're ready, unpause it and we'll go through it together. If you're struggling, I'm going to go through it step by step really slowly so maybe leave the video running for a little bit longer, just to see me just start filing in the frequency tree, when you're ready, you can pause it and finish. You want to start now, pause in three, two, one. So the first step is to fill in this first one. Three students passed despite not doing their homework so they didn't do their homework and they passed, that's three students. That one's rather easy to fill in. Start the after that. Half of the group did their homework, we don't know how many people are in the group, we got nearly enough information on there yet, now to just leave that one to one side at the moment, and let's go on to the next bit. 8 of those who did their homework failed the test. So they did their homework and they failed, that's eight. That's going to go there. Okay and tick that, there's nothing more I can use it for yet. Twice that number passed the test all together, so we're referring to that eight there so that means it's 16 people passed all together. Now we know already that three people, didn't do their homework and passed. If 16 people passed all together, that means that 13 people must have done their homework, and passed the test. Now we can start to work backwards here. If 13 people did it and passed, eight people did it and failed, that means that 21 people did their homework. Now we know, we go back here, that half of the group did their homework. That means half of them didn't. So that means we've also got 21 students there, so in total, there must be 42 students. Now the last thing to fill in, is how many students didn't do their homework and then failed the test. And if 21 students didn't do their homework, three of them passed that means that 18 failed. Now we can check this, because at each stage, we should be getting a total of 42, we know there are 42 students altogether. That they did their homework or they didn't. Now again, we've got 42 students, that they passed or they failed, well 13 passed, add another three, 16 of them passed, eight of them failed, and another 18 failed, that makes 26 that failed, now we can add those together and total once again, we've got 42. So we can be very confident that we got this one right. That's it for today, I'll see you later. Goodbye!

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

In this lesson, we will look at how we can represent combined events as frequency trees.

Maths

Y9 Maths

Key Stage 3

Year 9

Frequency trees

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)

Calculating the probability of combined events

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"?

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

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:Frequency trees

Core Content

Probability: