Lesson video

Hello and thank you for choosing this lesson.

My name is Dr.

Ronson and I'm excited to be helping you with your learning today, let's get started.

Welcome to today's lesson from the unit of probability and theoretical probabilities.

This lesson is called the Probability Scale, and by the end of today's lesson we'll be able to assign a numerical value to an outcome to give the probability that the outcome will occur.

Here are some previous keywords that we're going to use again during today's lesson.

So if you need to remind yourselves what any of these words mean, you may wanna pause the video while you do that, and then press play when you're ready to continue.

And this lesson will also introduce a new keyword, and that is the word probability.

The probability that an event will occur is the proportion of times that the event is expected to happen in a suitably large experiment.

Now if that's not clear straight away, don't worry about it.

We'll see plenty of examples of this during today's lesson.

The lesson will contain two learning cycles.

In the first learning cycle, we'll be looking at probabilities that are expressed as percentages and as decimals.

And then in the second learning cycle, we'll looking at probabilities that are expressed as fractions.

But let's start off with expressing probabilities with percentages and decimals.

Here we have Alex and Alex has a spinner.

Now this spinner looks quite nice because both of the sectors say win on it.

He's going to spin the spinner several times and how many times should he expect to win? Well, he says, "Winning is certain each time because it's the only possible outcome.

So I'll win all of the time with this spinner." So here we have a different spinner, which doesn't quite look so nice.

Both sectors say lose.

He's gonna spin this one several times, how many times should he expect to win? Alex says, "Winning is impossible here because it is not one of the possible outcomes.

So I'll win 0 times with this spinner." So here we have another spinner, which seems a bit more reasonable.

One half says win, the other half says lose.

Alex is gonna spin this several times.

How many times she expect to win? Alex says, "The two outcomes both have an even chance, so I should expect a similar number of each outcome.

That means I'll probably win around half of the spins." He also adds, "That might not necessarily be what happens." He might win more than half the time or less than half the time.

It's just the most reasonable result to expect based on how this spinner looks.

So let's be a bit more specific now.

If Alex spins this spinner 100 times, approximately how many times should he expect to win? Alex says, "I should expect around 50% of the spins to be of each outcome.

That means I'll probably win on around 50 out of the 100 spins." He also adds as well that, "That might not necessarily be what happens." He might get 51 wins, he might get 60, 70, more, or even fewer than 50 wins.

But it just seems the most reasonable result to expect based on how this spinner looks.

So what if Alex spins the spinner 20 times? What should he expect to happen? Alex thinks, "I'd still expect around 50% of the spins to be of each outcome.

That means I'll probably win on around 10 outta the 20 spins." So what if he spins a spinner two times? What should he expect to happen this time? Alex says "There are two spins and two equally likely outcomes.

So I'd expect one of each outcome.

That means I'll probably win on one out of the two spins." Now that might not necessarily happen.

You might win twice, you might lose twice.

But it seems most reasonable to expect one of each with these two outcomes being equally likely.

But what if he spins the spinner just once? What should he expect to happen this time? Alex says, "Either outcome could happen, so I don't know what to expect." He knows that he's got an equal chance of getting a win or lose, but doesn't know what to expect in that one spin.

This brings us to probability.

Probability is a measure of the likelihood of an outcome happening.

Words such as like and unlikely are words that describe the likelihood of an event, but they are relatively vague.

Probabilities describe the likelihoods of an event in a more precise way by using a numerical value to indicate its likelihood.

The probability that an event will occur is the proportion of times that the event is expected to happen in a suitably large experiment.

We've seen a bit of that with Alex so far where he said, "I expect around 50% of the time I will get a win." That's kind of what we're talking about here with probability.

So here we have a scale of likelihoods and let's now turn this into a probability scale.

And let's start off with what happens when an event is certain to happen.

An event that is certain has a 100% probability of happening.

For example, with this spinner, no matter how many times I spin it, I know that 100% of the trials conducted this spinner would be a win.

So that's why the probability is 100%.

An event that is impossible has a 0% probability of happening.

So for example, with this spinner, no matter how many times they spin it, 0 of the trials would result in a win.

So 0% is the probability relates to impossible.

An event that has an even chance has a 50% probability of happening.

For example, with this spinner here, if we conduct a large number of trials with this spinner, we'd expect that approximately 50% of the outcomes would be a win.

Now, that might not necessarily be exactly what happens.

We might get more wins and losers and more losers and wins, but we should probably expect around 50% of the time to get each outcome.

And the more trials we do, the more likely it is that that proportion would get to around about 50%.

Probability is greater than 50% have more than an even chance of happening.

So probability such as 60%, 70%, 75%, and so on, these indicate that an event is likely to happen.

For example, if we conducted a large number of trials with this spinner, then we would expect it to land on win more often than it lands on lose.

Probabilities less than 50% have less than an even chance of happening.

So that means they are unlikely to happen if they have a probability of around 10%, 20%, 30%, 40% or any number in between 0 and 50%.

For example, with this spinner here, if we conducted a large number of trials with this spinner, we would expect it to land on win less often than it lands on lose.

So now we have a probability scale where the probabilities are expressed as percentages ranging from 0% being impossible to 100% being certain.

Probabilities can also be expressed using the equivalent decimals between 0 and one, where a probability of 0 means that an event is impossible and a probability of one means that an event is certain.

And remember that one whole is equivalent to 100%.

And the size of the number indicates how likely it is for an event to occur.

The greater the number between 0 and one, then the more likely it is for that event will happen.

So let's check what we've learned.

Which likelihood describes an event with a probability of 100%? Is it A, impossible B, unlikely C, even chance, D, likely or E, certain? Pause the video while you make a choice and press play when you're ready for an answer.

The answer is E, certain.

Which likelihood describes an event with a probability of 0.

7? Same options again.

Pause the video while you make a choice and press play.

when you're ready for an answer.

The answer is D, likely.

0.

7 is greater than 0.

5, which means it has a likely chance of happening.

Which probabilities suggest that an event is unlikely.

You've got a few to choose from here and there are multiple correct answers.

Pause the video while you make your choices and press play when you're ready for some answers.

So probabilities that are unlikely here, are these ones, 0.

1 and 0.

48 and 0.

235 are all less than 0.

5 when they're expressed as decimals.

And 1% and 18% are both less than 50% expressed as percentages.

So these are all less than even chance, they're all unlikely.

And here we have three spinners.

One of these spinners has a 48% probability of landing on win, which spinner is it A, B, or C? Pause a video while I make a choice and press play when you're ready for an answer.

The answer is B.

The fact it has 48% probability means it is less likely to land on win than it is to not land on win.

Now I can see that with both A and B, but with A, this sector for win is so small that that is a very small probability.

Where the one for B, this sector's just less than halfway and that is also the case at 48% as well.

It's just less than 50%.

Here we have a scenario where a spinner has four outcomes, A, B, C, and D.

The outcomes are not equally likely to happen on this spinner.

The probability that the spinner lands on A is 25%.

The probability of the spinner landing on B is 18%.

The probability of the spinner landing on C is 34%.

And the probability of the spinner landing on D is 23%.

That was a mouthful.

Sam agrees as well.

Sam says, "It seems like a long way of stating these probabilities.

I wonder if there's any notation that could be used to shorten this information rather than writing out these long sentences each time." Well the good news, Sam, is there is.

Statements about probability can be expressed more efficiently by using the notation P and in your brackets, your outcome and then equals.

For example, this sentence here, the probability of the spinner landing A is 25% can be written like this.

Where P stands for probability.

What we have in our brackets is the outcome.

So that first bit we've indicated there is the first part of that sentence, the of the spinner landing on A, and then you have equals and the numerical value of the probability, which in this case is 25%.

So back to our scenario with the spinner, again, the spinner has four outcomes, A, B, C, and D, and here are the probabilities for each outcome.

Now, if we were to conduct a large number of trials of this spinner, which outcome would you expect to occur the most often? Well, based on these probabilities, we'd expect outcomes C to occur most often because it has the greatest probability, so we'd expect the greatest proportion of outcomes to be for C.

Which outcome would you expect to occur the least often? Well, in this case, we'd expect outcome B to occur the least often.

It has the lowest probability, so we'd expect it to have the smallest proportion in terms of the number of outcomes in a suitably large experiment.

The spinner could look a bit like this or it could have multiple sectors for C and multiple sectors for each letter.

But in terms of the proportion of the spinner that is taken up by each letter, it would be similar to what we can see here.

So if there were 2000 trials of this spinner, approximately how many times could the spinner be expected to land on A? Now we can't predict exactly how many times it'll land on A, but we can use the probability to get a good sense of what we might expect.

The probability it lands on A is 25%, and that means we should expect about 25% of the trials to result in A as its outcome.

25% of 2000 is 500.

So that makes a good reasonable approximation for how many times we expect it to land an A in 2000 trials.

It might not do, but that seems like the most reasonable approximation to make based on the probability.

So let's check what we've learned.

A video game based on chance has a 50% probability of a win.

Which number below is the most reasonable estimate for how many wins could be expected in 1000 games? You've got four options to choose from.

Pause the video while make a choice and press play when you're ready for an answer.

Well, there are 1000 games and would expect 50% of those roughly to result in a win that is 500.

The same video game again has been adjusted now, so it has a 25% probability of win, which is now the most reasonable estimate for how many times to expect a win in 1000 games this time.

Pause the video while you make a choice and press play when you're ready for an answer.

Well this time we're expecting roughly 25% of the games to be a win and that would be 250.

Okay, it's over to you now for task A.

This task contains three questions and here are questions one and two.

In question one, you've got the scale of likelihoods.

You need to write down the probabilities on to that scale in the correct order.

If you can put 'em in roughly the right places, great, but just make sure they definitely are in the right order on that scale in approximately the right places.

And in question two, you gotta do the same thing again, but this time the probabilities are expressed as decimals.

Pause the video while I do these and then press play when you're ready for question three.

And here is question three.

In each question you're given a spinner and a choice of four probabilities.

In each question, tick the probability that the spinner would land on win.

Pause the video while you do this and press play when you're ready for some answers.

Right, let's go through some answers to questions one and two.

In question one, we need to put these probabilities on the scale.

They should go here.

They should be in this order and hopefully you've got 0, 50% and 100% above those lines and you've got 20% and 45% in the unlikely zone and 55% and 75% in the likely zone.

The exact positions of those don't worry about it.

If you have used a ruler to do it accurately, brilliant, but otherwise, if they're in the right place, I'm happy with that.

And in question two, same again with the decimals.

They would go here, 0, 0.

5 and 1 will be above those lines there.

0.

1 or 0.

35 will be in the unlikely section of the scale.

0.

6 and 0.

95 will be in the likely section and if you've got 'em roughly in these places, brilliant.

And now let's go for the answers to question three.

We to choose the right probability for getting a win in each spinner.

In part A, we can see that the win is an even chance, that's 50%.

In part B, you are certain to win with that spinner, so that probability is one.

If you put 100, you may be getting mixed up with percentages there.

These probabilities are not expressed as percentages, so 1 is the greatest probability it could have.

In part C, we can see that you are unlikely to win with this spinner because the win section is smaller than the lose section.

There are two probabilities there that are unlikely, 0.

1 and 0.

45.

0.

45 is just less than 0.

5.

Where we can see a sector is really small for win, so it's gonna be 0.

1.

With part D, you can see that the sector for win is just slightly more than half of the spinner, which means you are more likely to win than not win.

But it's not as much as 95%, it's only a little bit more than 50%, it's 52%.

With part E you can see that you've got a likely chance of winning.

It's not quite as much as 98%, which is nearly certain.

It's quite a bit more than 52%, which is just over a 50% chance, even chance, it's 75%.

And in part F, the probability of winning this time, it's just a bit less than 0.

5, it's just a bit less than an even chance, it's 0.

45.

Great work so far.

Now let's do it again with fractions.

Probabilities can also be expressed using the equivalent fractions.

So here we have our scale of likelihoods.

Let's now turn this into a probability scale using infractions.

A probability of 0 means an event is impossible.

Just like with decimals and percentages.

A probability of 1 whole means that an event is certain, just like with decimals.

And also let's remember that 100% is the same as 1 whole as a fraction or decimal.

A probability of a half which is equivalent to 50% are 0.

5.

A probability of a half means that an event has an even chance.

And probabilities that are less than a half indicate that an event is unlikely to happen.

And a probability that are greater than a half indicate that an event is likely to happen.

So let's take a look at an example of this in action.

Here we have Andeep, Izzy and Laura, who are playing a board game that involves rolling a regular six-sided dice.

They've each noted which numbers would help them on their next dice roll.

Let's hear what they say.

Andeep says, "The probability I'll roll a number I want on my next turn is three sixths." Izzy says, "The probability that I'll roll a number on my next turn is one sixth." And Laura says, "The probability that I'll roll a number I want on my next turn is four sixths." So who is the most likely to roll a number that they want on the next turn? Let's remember that the greater the probability, the more likely it is an event will happen.

So what we're really trying to figure out here is out of these three probabilities, which are expressed as fractions, which has the greatest value? Well in this case, Laura is the most likely to get a number she wants because four six is greater than one sixth and also three sixths as well.

Who has an even chance of rolling a number they want? Let's remember that even chance is the same as 50%, 0.

5 or one half depending on how you are expressing your probability.

So which of these fractions is equivalent to one half? That would be Andeep.

Three six is equivalent to one half.

So Andeep has an even chance of rolling the number he wants.

Blank is three times as likely as blank to get a number he wants.

Who is three times likely as who? Well Andeep is three times as likely as Izzy to get a number he wants because three six is three times one six.

Let's look at a different scenario.

Aisha, Jacob and Sophia are playing a game where each person uses a different spinner to determine if they gain points.

Let's hear from each of them.

Aisha says, "The probability I gain points of my spinner is one quarter." Jacob says, the probability that he gains points with his spinner is one eighth.

And Sophia says, the probability that she gains points with her spinner is three eighths.

Out of these three people who is the most likely to gain points with their spinner? Well, let's remember about the greater the probability, the more likely it is an event will occur.

So we're trying to figure out out of these three probabilities, which again are expressed as fractions, which is the greatest? When comparing fractions with different denominators, it can be helpful sometimes to change so the denominators are the same.

So one quarter is the same as two eighths, and that can be compared to one eighth and three eighths.

That means Sophia has the greatest probability of gaining points with her spinner.

Who is the least likely to gain points of their spinner? Well, that would be Jacob.

One eighth is smaller than a quarter and it's also smaller than three eighths.

So let's check what we've learned.

The probabilities for three unrelated events are listed below.

Which event is least likely to happen? Is it A, B or C? Pause the video while you make a choice and press play when you're ready for an answer.

The answer is A, five tenths is the smallest of those three probabilities.

Which event is certain to happen? Is it A, B or C? Pause the video while you make a choice and press play when you're ready for an answer.

The answer is C.

10 tenths is equivalent to 1 whole or 100%.

So that means it's certain.

Here we have three unrelated events, again, D, E, and F and their probabilities.

Which event has an even chance of happening here? Is it D, E or F? Pause the video while I make a choice and press play when you're ready for an answer.

The answer is E, four eighths is equivalent to one half and that's an even chance.

Which event is most likely to happen this time? Pause the video while you make a choice and press play when you're ready for an answer.

The answer is D, four sevenths is the greatest of those fractions.

Now you could, if you want to get a common denominator between sevens, eighths and ninths, we don't have to in this case because the numerator are all the same.

So you can just do it by comparing denominators.

The one with the smallest denominator means it has the greatest value.

That's four sevenths.

Here we have Alex.

Alex has a spinner with four equally size sectors where three of them say lose and one says win.

Alex is going to spin the spinner 40 times and record his results.

How many times should he expect the spinner to land on win? In other words, how should Alex finish this sentence? He says, "I expect the spinner to land on win approximately blank times." Pause the video while you think about this and press play when you're ready for an answer.

Alex says, "I expect a spinner to land on win approximately 10 times.

That's because I would expect it to land on each sector an equal number of times, but that might not necessarily happen." It is just the most reasonable thing to expect.

Let's think about expectation a little bit more.

An outcome with a probability of one quarter could be expected to happen in approximately one quarter of the total trials performed.

That's what the probability tells us.

For example, in an experiment with 40 trials, we might expect the outcome to happen approximately 10 times.

That's one quarter of 40 and that might not necessarily be what happens, but it just seems the most reasonable thing to expect based on its probability.

But the more trials an experiment contains, the more likely it is that the proportion of outcomes will be similar to what was expected.

So let's check what we've learned with that.

A spinner has four numbers on it.

The probabilities of the spinner landing on each number are listed below.

During a game, the spinner has spun a total of 100 times which number could be expected to occur most frequently out of these four outcomes? Pause the video while you make a choice between 1, 2, 3, and 4 and press play when you're ready for an answer.

The answer is two.

That's because the probability for two is the greatest of all four of those probabilities.

Approximately how many times could the spinner be expected to land on the number four? Pause the video while make a choice and press play when you're ready for an answer.

Well, the probability is one quarter.

We'd expect it to land approximately one quarter of the time out of those 100 spins, that would be 25 times.

Well, it's over to you now for task B, this task contains four questions and here are questions one and two.

Each of these questions you have a scale of likelihoods and some probabilities that are expressed as fractions.

You need to place the probabilities in the correct order on this scale, aiming to get each fraction in the correct zone on the scale of likelihoods there as well.

Pause the video while have a go at these and press play when you're ready for question three.

And here is question three.

Here we have a table with some information about some events.

The first column tells you the event.

The next column expresses the probability when it happens as a fraction.

The next one is as a decimal and then as a percentage.

The next column describes the likelihood that it happens such as with words as unlikely, and even chance.

And the last column is about if we had 600 trials, how many times we expect that event to occur.

So for example, in the top row there we have a probability of one half which is equivalent to 0.

5 and 50%.

That's an even chance.

So if we did 600 trials, we'd expect that example event to occur 300 times.

What I'd like you to do, please, for events A, B, C, and D is fill in the gaps on the table.

Pause the video while you do that and press play when you're ready for question four.

And here is question four.

A dropped cone could land in one of two ways, could land based down or curve down.

A large experiment concluded the following probabilities, the probability it lands based down is three tenths.

The probability it lands curve down is seven tenths.

And you have some questions to answer based on those probabilities.

Pause the video while you do this and press play when you're ready to go through some answers.

Okay, let's now go through some answers.

In question one, our fractions go here.

We have one 20th, which is not over the impossible, but it's just in the unlikely zone.

Then we have five twentieths, nine twentieths, we don't have anything above even chance.

We have then 11 twentieths in the likely section and then 18 twentieths and then we have 20 20th should be over certain.

Question two, here are our fractions.

Again, there is nothing over impossible.

We have five one thousandths and five elevens in the unlikely section.

Five tenths is equal to an even chance.

Then we have five ninths and five sixths in the likely section and five fifths is equal to certain.

And then question three, event A has a probability of 0.

25, which is equivalent to one quarter or 25%.

If we had 600 trials where A is a possible outcome, then we'd expect it to happen 150 times.

For event B, the probability is 75%, which is equivalent to three quarters and 0.

75.

That means it's likely to happen.

And if there were 600 trials where B was a possible outcome, we'd expect approximately 450 of them to be B.

And event C, the probability is two fifths.

That's equivalent to note 0.

4 or 40%.

It means the event is unlikely to happen.

Therefore, if we had 600 trials where C was a possibility, we'd expect it to happen approximately 240 times.

And then with event D, why would we expect 0 trials? That's because it's impossible.

That means it's got probability of 0 or 0 or 0% depending on how you expressing it.

And then part C, if you were able to use the link to access the cone simulation, you would've ran a hundred trials and then compare the results from your simulation to your prediction from part B to see how similar they are.

Now your answers here may vary depending on what happened in your simulation, but here are some example answers you could give.

You could say, for example, the results were similar to my prediction but not the same.

Another example answer could be there were 89 results for curve down, which is more than expected.

That number 89 might be different for you depending on what your simulation did.

Or the number of times it landed curved down was much more than the number of times it landed based down, which is what I expected, but the exact number of times it landed on each side was different to what I expected.

Something like that may be appropriate for your answer of part C.

Great job during today's lesson.

Let's now summarise what we've learned.

Probability is a measure of the likelihood of an outcome happening.

And probabilities take numerical values between 0 and 1, but they can be expressed in different ways as percentages, as decimals, as fractions for example.

A probability of 0 means the outcome is impossible.

A probability of 1 or something that's equivalent to 1 means an outcome is certain.

And the probability of an event will occur is the proportion of times the event is expected to happen in a suitably large experiment.

Thank you very much.

Well done.