When you roll a fair six-sided dice, it is harder to roll a six than a four.

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

There is an even chance of it raining on any given day.

If I roll a fair six-sided dice, I am more likely to get an even number than an odd number.

Since a football team can either win, lose, or draw, the probability that they win must be one third.

Representing Probability

﻿Hello. I'm Mr. Langton. And today we're going to look at complimentary events. I have a haircut. Thank you for noticing but about those sorts of compliments. Today we're looking at probability. 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. So we'll start off with the try this activity, decide on a way to arrange the statements into groups. How would you describe the groups? Which statements could you write that could fall into each group? Pause the video and have it go when you're ready and unpause it and we go through it together. Pausing in three, two, one. So how did you get ? I've arranged them into two groups. Three of them are coloured in pink and the other five are coloured in green. Now in each case for every single one of the eight options you're being given a choice between two things which are the heads or tails, it's a three or it's a six, it's a weekend or a weekday. Every single time you're choosing between two different things. The ones that are coloured in pink, you can only have one of those two things happen. If you flip a coin it can only be heads or tails. If you pick a day at random on a calendar it must be a weekday or weekend. As for the green ones if you roll the dice you could get a three or a six but you could also get a one or two or four or five. If we look at the weather it could be sunny or it could be rainy or it could be snowy it could be cloudy it could be windy there are a lot more options. So that's how I split them into my two different groups. The pink ones are the ones where there are only those two things that can happen. And the green ones are the ones where at least those two things could happen and other things as well. You may be able to think of some of your own statements that would match those. So complimentary events are those were only two outcomes are possible. Either something happens or it doesn't it must be this one or it must be that one. There's no third or fourth option. So we picked two of them here. The first one the probability to getting a head or a tail when flipping a coin. Probability to getting your head is a half. And the probability of getting a tail is a half. And the total of those probabilities if I have them both up together total is one. Now if I pick a random day on a calendar what's the probability that it's a weekday when there are five weekdays on that Monday, Tuesday, Wednesday, Thursday, Friday. That's my five weekdays. And there's also Saturday and Sunday. So it's 5 out of 7. What's the probability of not picking a weekday? Well that's these two Saturday and Sunday. So that's going to be two out of seven. And again what's the total of the probabilities of five sevenths add two sevenths is seven sevenths, but once again a whole one. And that's the really important thing to note about complimentary events. We said that they're the only two things that can happen is that the one of them or the other and the two probabilities of those two things will always add up to make one. So if we look at this last example, there are 20 coins in a bag. Six of them are one pound coins. Without looking I take a coin from the bag. What's the probability that is not a one pound coin? Well, there are six one pound coins and there are 20 coins altogether. So that's going to be simplified out to three tenths. The probability that I do get to one pound coin is three tenths. So because these are complimentary events I either get a one pound coin or I don't, that means if the probability of getting the coin is three tenths the probability of not getting it is going to be one takeaway three tenths which is seven tenths because there's two probabilities three tenths and seven tenths must add up to make a whole one. Okay, now it's time for you to have a go. Pause the video and access the worksheet when you're done when you unpause it we can go through it together. Good luck. So how did we get on, let's go through the answers together. The probability that Anthoni passes his driving test is 0.6. So what is the probability that he fails? That would be 1 subtract 0.6 so 0.4. Next up, there are 15 marbles in a bag, four of them are red what is the probability of picking one which is not red? So the probability of getting red is four elevenths. I need to take that away from one. And that's going to give me seven elevenths. And that seven elevenths. If there is a 30% chance to rain on Sunday then what is the probability that it will be sunny? Now, some of you might've thought it'd be 70% but that's not true. The actual answer is that we don't know it's going to be less than 70% or up to 70% it can't be bigger than that but the probability of just rain and just sunny those two events are not complimentary on their own there are more options that can happen. So we can't say for certain what the probability will be. That was a trick question little bit sneaky. Now move on to the fourth one Xavier has got a biassed coin. It has an 83% chance of landing on heads. What's the probability that it lands on tails? So is a 100% chance altogether take away the 83% and that's going to leave us with 17%. Move on to the last one. Cala has some round and square counters. She picks one at random what is the probability that she does not choose a square one? And we've got to choose all the correct answers from this list. So the ones that are not square that's going to be one, two, three, four, five, six out of 10 Probability of getting one that's not square is 6 out of 10. So we've got 6 out of 10 there. I'm off to a good start. 6 out of 10 is the same as 0.6 which is good it's obviously not the same as 4 out of 10 So you may cross that one off. And 6 out of 10 will cancel down to three fifths. And if it's not 0.6 and it can't also be not 0.4 and it can't be not 0.3. It's not going to be 40%. Now, there are six and, sorry, six are circles and four of them are squares. So what's the ratio in circles to squares is six to four. We can't have a probability as a ratio that's not what ratios are for. So that is not one of our answers. Finally, we've got the explorer activity. What is the probability of choosing a green cube from each of the bags? Next up, Cala has six more cubes: four are green and two are white. She decides to share all six of her cubes into the three bags. What different sharing strategies could she use so that either green is most likely from bag A, green is most likely from bag C or green is least likely from bag B. Pause the video and have a go, when you're ready unpause it we'll have a look at it together. And pause in three, two, one, go. So let's go through the answers. On the starts off the probability to getting a green one from each bag. From the first one is going to be two fifths. And the second one it's going to be four sevenths. And the third one is going to be one fifth. Now I can't write in white because you won't be able to see it. But the probability of getting a white cube is going to be three fifths because together it should make a whole one. So the probability to here it's going to be three sevenths that make a whole one and that's going to be four fifths. So that's my probabilities that's part Aidan. Now, Cala's got six more cubes four of them are white. Sorry, four of them are green and two of them are white. She's going to share the six cubes out but try to and make each of these possible scenarios happen. So the first one making green most likely in bag A. Well, I'm going to put my four greens into this bag. And that now makes the probability of selecting a green from in there six ninths. And there are currently seven cubes in bag B, if I were to add the two white ones there then that's going to be there's now nine cubes in there which means that there are only four ninths of green. And I know that six ninths is bigger than four ninths so that's a strategy that works. So let's look at the next one to make green most likely in bag C we going to have to put the greens in there also. If I put four greens in there and now the probability of getting a green is 5 out of 9. I'm going to have to put once again both of my whites into B because that makes the probability of getting a green there 4 out of 9 and five ninths is bigger than four ninths. So that works for that one as well. Finally, making green the least likely option in bag B. This one is quite a tricky one. You're going to need to put one green in A, which makes that probability 3 out of 6 or a half. You now have to put the other three greens by . Now to put the other greens here which makes that one two that makes up 4 out of 8, Doesn't it? Which is also a half. So at the moment the probability of eight in C are both equal are both a half. Now at the moment B has got four sevenths green so that's more that's too many that's more than half but we still got two white cubes. If we put our two white cubes in here and the probability of getting a green now is four ninths which is less than a half. So we've done it. That's it for this lesson I'll see you later. Goodbye.

There are 20 guests at a party. 12 of them are drinking coffee and the rest are drinking tea. What is the probability of selecting one who is drinking tea?

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

The probability that it rains tomorrow is 0.4. What is the probability it is sunny?

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?

The probability that it will snow tomorrow is 0.05. What is the probability that it won't snow?

Complementary Events

In this lesson, we will look at how we can calculate probabilities for complementary events.

Maths

Y9 Maths

Key Stage 3

Year 9

Complementary events

Probability

Statistics and probability

THEME

UNIT

Representing probability

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

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?

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:Complementary events

Core Content

Probability: