Lesson Video: Probability: Certain, Possible, Impossible Mathematics

In this video, we will learn how to use the words “certain,” “possible,” and “impossible” to describe the probability of an event occurring.

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Video Transcript

Probability: Certain, Possible, Impossible

In this video, we’re going to learn how to use the words “certain,” “possible,” and “impossible” to describe the probability or the chance of different things happening.

To begin with, let’s go over our three words, and these are words we use in everyday life all the time. If we say that something is certain, we’re saying that it’s definitely going to happen. There’s no doubt about it. Sometimes, it’s hard to think of things that are definite or certain. A good place to start though is maths. There are lots of certainties in maths.

For example, if we have a square and we measure the angle of one of its corners, we know that we’re going to find that it will measure 90 degrees or a right angle. It always will. This is one of the things we know makes a square a square. It has four corners and they’re all right angles. So we could say that it’s certain that the angle inside the corner of a square will measure 90 degrees. And maths has lots of other certainties too.

Now, it might be interesting if we think about the word impossible next. And the reason is this is the complete opposite to something being certain. We use the word impossible to describe something that definitely won’t happen. It’s never going to happen. There’s no chance at all of it happening. These are the examples that is always quite fun to think of. If you press pause on the video for a moment and peek underneath your chair, it’s impossible that you’re going to see a snowboarding polar bear down there. There’s no chance of it happening. It’s not even slightly possible, which brings us onto our third word “possible.”

If certain is something that’s definitely going to happen and impossible is something that’s definitely not going to happen, then we can think of the word possible as living in between these two opposites really. In maths, we use the word possible to describe something that might happen. There’s some chance that it might happen. We could say it’s possible that it will rain tomorrow. We couldn’t say it was certain, even if we lived in a really wet place. And we can’t say it’s impossible, even if we lived in a really hot place. That would mean there was no chance at all of it happening.

Instead, we say it’s possible. There’s a chance that it might happen. Might not be very big chance, but there’s a chance. Now that we’ve learned what these three words mean, let’s have a look at some more examples and see whether we can say whether they’re certain, possible, or impossible.

Is this event certain, possible, or impossible? A goat will fly past the window.

Now, it’s not very often in maths that we get a question about flying goats, is it? But we’ve got one here, and it’s all about probability. The chance of something happening. And the event that this question is talking about is the chance that a goat is going to fly past the window. Our question is asking us if we were to get up from our computer now and look out of the window, what’s the chance that if we look up to the sky, we’ll see a goat flying past?

Now, we could answer this question in lots of different ways, but we’re given three different words to choose from. We need to answer using one of them. Is this event certain? Is it possible or impossible? Let’s remind ourselves what these three words mean. We use the word “certain” to describe something that definitely will happen. There’s no doubt about it. There’s no chance at all of it not happening. It’s certain. The opposite of certain is actually our third word, “impossible.” If an event is impossible, it’s definitely not going to happen. There’s no chance at all of it ever happening. It’s absolutely impossible.

But then, in between these two opposites, we’ve got the word “possible.” And we use this word for events that could happen. They might happen. They might not, but there’s some chance that they could. Now, which one of our three words are we going to use to describe a goat flying past the window? Well, hopefully the idea of a goat flying made you smile. It’s not possible at all, is it? Goats don’t fly. We know it’s not certain, but it’s also not even possible. There’s not the slightest chance it’s going to happen; it’s impossible. We can say that the probability or the chance that a goat will fly past the window can be described as impossible.

Is it possible, impossible, or certain that someone watches television four times a week?

This question is all about the likelihood of something happening. Is it possible that it will happen? Is it completely impossible? Or is it absolutely certain? We can use these words to describe the probability of something happening. And the event that this question is talking about is someone watching television four times a week. Let’s have a think about our three possible answers. If it’s certain that someone watches television four times a week, then it’s always true that they watch television four times every week, never any less, never anymore.

If we say that something’s impossible, we’re saying that there’s no chance of it happening. It’s not even possible that someone could watch television four times a week. And then we’ve got the word “possible.” If a statement like this is possible, then there’s a chance that it could happen. It might not happen, but there’s a chance that it could. I wonder, how many times do you watch television a week? Although some people might watch television four times a week, we can’t say that it’s always true, can we? But we can’t say that it’s never true, either. When something’s impossible, there’s no chance of it happening.

Of course, there is some possibility that someone could watch television four times a week. It’s quite easy to do really, isn’t it? The only word we can really use to describe this event is “possible.” It’s possible that someone watches television four times a week, and it’s possible that they don’t. It’s a statement that’s sometimes true. Maybe it’s true of you. Maybe it’s not. It’s possible that someone watches television four times a week.

Now, so far in this video, we’ve had to think about events that happen or don’t happen in the case of flying goats in everyday life. And we looked at just one event every time. What’s the probability of it raining? What’s the chance of there being a snowboarding polar bear? But, you know, sometimes we have to apply some more maths than this. We could start using the words more and less. Is something more likely to happen than something else or less likely?

Let’s use this empty plate for a taste test. And let’s imagine there are three types of sweets. There’s the toffee twirl and the strawberry supreme, both of which are delicious. But then we’ve got these orange-colored sweets, coffee and carrot crunch. Let’s imagine that we really don’t like the sound of these, and we want to avoid them as much as possible. But just to make things really difficult, we’re going to do this taste test with our eyes covered up. Here it goes. Now thankfully, on the screen here, we’ve got the benefit of being able to see the plate of sweets. What do you notice?

All of the sweets are that flavor we don’t like, aren’t they? So if we were to come back to our three words again, we could use the word “certain.” It’s certain that a coffee and carrot crunch is going to be chosen. There’s no doubt about it. It’s definitely going to happen. It doesn’t matter which sweet we pick. We’re definitely going to have to eat a coffee and carrot crunch. And because this is certain, we could say that picking one of the other two flavors is actually impossible. There’s no chance of it happening.

Now, let’s change our plate of sweets. This is interesting. This time we’ve got five strawberry supremes and two toffee twirls. And we have to say we’re very grateful that there’s not a coffee and carrot crunch to be seen. So what can we say about the possibility that we’re going to pick one of those horrible flavored sweets now. Well, there aren’t any to pick. So it doesn’t matter which sweet we choose; we’re never ever going to have to taste the coffee and carrot crunch. It’s impossible.

But can we use the word “certain” to describe anything this time? Well, I suppose we could say it’s certain that we’re going to get a flavor that we like. But because there’s a mixture of flavors this time, we can’t quite be sure which one we’re going to end up with. It’s possible that we’ll pick a toffee twirl. But then there are some strawberry supremes on the plate too. And it’s possible that we could pick one of those. Can you see how we had used the word possible here to describe the chance of us picking one of these two flavors?

But you know, there’s something else we could say about these sweets. And it comes back to what we were saying a few moments ago about using the maths words more and less. If we look at this plate of sweets, we can see that there are more red sweets than there are blue, can’t we? So if we were going to choose one of these sweets at random — in other words, without seeing it, without planning what we wanted to pick — we could say that it’s more likely that a red sweet is going to be picked than a blue sweet. It’s possible that we’ll pick a blue sweet, but it’s more likely that we’ll end up with a red sweet.

Even without seeing what we’re doing as we put our hand on the plate, it’s more likely that we’re going to pick a strawberry supreme. There are more of them. And of course, the opposite’s true. It’s less likely that a blue sweet is going to be chosen than a red sweet. But don’t forget, it’s even less likely that we’re going to pick a coffee sweet. It’s impossible. So as well as thinking about real-life events like we did earlier on, we can also use the words ”certain,” ”possible,” and ”impossible” in sort of experiments like this, where we pick things randomly. Let’s have a go at answering a question that’s a little bit like this sweet example. And it involves choosing something else that we can’t see.

If there are three red balls and nothing else in a bag, is it certain, possible, or impossible that if I pick out a ball at random, it will be red?

This is quite a wordy question really. So let’s go through it really slowly to understand what it’s asking. Firstly, we’ve got a description and that’s of three red balls and nothing else in a bag. Let’s sketch them. Here are our three balls. And let’s use this circle to represent the bag that they’re in. Now, there’s a really important part of that first phrase that we haven’t mentioned yet. And, it’s these two words here. We’re told that there are three red balls and nothing else in the bag. There aren’t any other colors. There aren’t any other objects. Those three red balls are the only things in that bag.

Now this question is all about what’s gonna happen if we pick out a ball at random. When something like this happens at random, it happens by chance, without any planning or decision-making. If we wanted to pick a ball at random, maybe we’d close our eyes or, like in the example of this question, we put the balls inside a bag where we can’t see them. This way, we can’t choose what we pick. Now the color that we’re asked about is red.

So this question is asking us, what’s the chance that we’re going to pick a red ball? And we’re given three possible answers. Is it certain that we pick a red ball? Is it possible that we pick a red ball? Or is it completely impossible that we pick a red ball? We know that the word ”impossible” means there’s no chance at all. It’s definitely not going to happen. Now, if we put our hand into that bag, we can’t say there’s no chance at all of us pulling out a red ball. We could say it’s impossible we’d pick out a green ball. It’s impossible we’re going to pull out a cheeseburger. But we can’t say it’s impossible we’re going to pull out a red ball. The bag’s full of them.

Let’s get rid of this as a possible answer. Speaking of possible, is it possible that we pull out a red ball? We know that when something is possible, there is a chance of it happening. It might happen; it could happen. Now this word sounds like it could be right. It is possible that we pull out a red ball. But it’s time to come back to those two words in that first phrase that we said were so important. There are three red balls and nothing else in a bag. We wouldn’t use the word possible to describe the chance of us pulling out a red ball. It’s much more than that. It’s absolutely certain because there’s nothing else in that bag.

If we put our hand in, wiggle it around, and pull out a ball, it’s going to be red, definitely. And if something is definitely going to happen, we use the word ”certain.” If you’re not sure about this, try it. If you don’t have colored balls, then try three coins that are the same. As long as you don’t put anything else in your bag, then you’ll see that the chance of you pulling out a red ball or one of your coins is certain. If there are three red balls and nothing else in a bag, the chance of us randomly picking out a ball and it being red is certain.

So what have we learned in this video? We’ve learned how to use the words ”certain,” “possible,” and “impossible” to describe the probability or chance of something happening.