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

This lesson is called Calculating Theoretical Probabilities From Lists With One Event, and by the end of the day's lesson, we'll be able to find theoretical probabilities from a list of possible outcomes for an event.

This lesson introduces a new keyword, and that is a theoretical probability.

A theoretical probability is a probability based on counting a number of desired outcomes from a sample space where all the individual outcomes are equally likely to happen.

This lesson contains two learn cycles.

In the first learn cycle, we're gonna learn how to find theoretical probabilities by using spinners as our tool for understanding what's going on.

And then the second learn cycle, we're gonna apply what we've learned to find theoretical probabilities in other scenarios.

But let's start off with finding probabilities or events with spinners.

Here we have a spinner with two sectors.

One says A, the other says B.

A single spin of this spinner has two possible outcomes, A or B.

These outcomes are equally likely to happen because the sectors of the spinner are of equal size.

So in theory, we would expect each outcome to occur an equal number of times in an experiment with a large number of trials.

Therefore, we can find theoretical probabilities for each outcome.

The probability that the spinner lands on A is one half.

This is because one out of the two possible outcomes are A, and those two outcomes are equally likely to happen.

With this probability being expressed as a fraction, we can see quite clearly where each part of this fraction comes from.

The numerator, which in this case is one, is the number of outcomes that are A, and the denominator, which in this case is two, is the total number of outcomes.

This means that in an experiment with a large number of trials, we would expect approximately one half of the outcomes to be A.

The probability that the spinner lands on B is also one half and that's for the same reason.

There is one outcome that is B, and there are two outcomes altogether.

Therefore, this spinner is based on the theory that the spinner would be expected to land on each sector an equal number of times.

However, if we run an experiment in practise, that might not necessarily happen, you might get more A's than B's or more B's than A's, but in theory, we would expect approximately half to be A and approximately half to be B because those are the theoretical probabilities.

And here we have a different spinner where there are three possible outcomes, A, B and C.

And what we can see from the spinner is those three outcomes are equally likely to happen.

Therefore, the probability that the spinner lands on A is 1/3.

This is because one out of the three possible outcomes are A.

In this case, in the fraction 1/3, one represents the number of outcomes that are A and the three represents the number of outcomes altogether.

So this would mean that in an experiment of a large number of trials we'd expect approximately 1/3 of the outcomes to be A.

So let's check what we've learned there.

Here we have a spinner with letters A to E on it, and we can see about the size of the sectors that each outcome is equally likely to happen.

So what is the probability that the spinner lands on the letter A? Pause the video while you write down an answer and press play when you're ready to continue.

The answer is 1/5.

That's because one of the sectors is A, and there are five sectors which each have a different outcome, so it's 1/5.

The spinner is going to be spun 1,000 times.

Which seems the most reasonable approximation for how many times a spinner would be expected to land on A? Is it A, 100, B, 200, or is it C, 500? Pause the video while you make a choice and press play when you're ready for an answer.

The answer is B, 200.

That is 1/5 of 1,000.

Here we have Andeep and Laura, and they have a spinner with five sectors on it, but two of them are labelled B, and three of them are labelled A.

Now those five sectors are equal size, but we can see here that there are more sectors of A on than there are with B.

Andeep and Laura are discussing the probability that the spinner lands on A in a single spin.

Andeep says, "I think the probability is one half because there are two outcomes, and one of them is A." Laura says, "That doesn't seem right because more than half of the spinner is A, so we'd expect more than half of the spins to land on A." I wonder what you think about this.

Maybe pause the video and have a think about what Andeep and Laura are saying and decide who do you agree with and what do you think the probability is? And then press play when you're ready to continue.

Theoretical probabilities can be found in scenarios where a trial has equally likely outcomes, but the likelihoods of the spinner landing on each letter here are not equal.

It is more likely to land on A than B.

So how can we get a theoretical probability out of this situation? Well, we need to remember that the sectors are of equal size.

It's just that there are more sectors labelled A than B.

So the likelihoods of the spinner landing on each sector are equal because they are all of equal size, which means by considering each sector as a separate outcome within this trial, we can find the theoretical probabilities for each letter.

In other words, there are three outcomes which are A and there are two outcomes which are B.

So the probability that the spinner lands on A is 3/5.

This is because three out of the five possible outcomes are A if we consider those three A's as being separate and those two B's as been separate.

So this probability of 3/5.

The three represents the number of outcomes that are A and the five represents the total number of outcomes.

And the probability that the spinner lands on B is 2/5.

This is because two out the five possible outcomes are B.

So here when we have our fraction of 2/5, we can see the two represents the number of outcomes that are B and the five represents the total number of outcomes.

In an experiment involving 500 trials, we'd expect the spinner to land approximately 300 times on A and 200 times on B based on these probabilities.

This is because with A, the probability is 3/5 and 3/5 of 500 is 300.

And with B, the probability is 2/5 and 2/5 of 500 is 200.

It's also worth just making sure that you've got the right amount of total trials there.

300 plus 200 is 500, and that's the number of times we spin the spinner.

Laura says, "It seems reasonable to expect the spinner to land on A more often than B because it takes up more space on the spinner.

However, the actual results may differ to this." It may not land exactly 300 times on A and 200 times on B.

It might even land more times on B than A by chance, but it seems reasonable to expect the spinner to land in A more than B and be approximately these proportions.

Probabilities can also be expressed using the equivalent decimals or percentages.

So here we have the probabilities as fractions, 3/5 and 2/5.

If we express them as decimals, we'd have 0.

6 and 0.

4.

If we expressed them as percentages, we'd have 60% and 40%.

So when we think again about how many times we might expect A to happen in 500 trials, we could either do 3/5 of 500, 0.

6 multiply by 500 or 60% of 500, they are all equivalent calculations.

So let's check what we've learned there.

Which spinner has a probability of 2/3 for landing on win? Your choices are A, B and C.

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

The answer is B.

On that particular spinner, we can see we have three equally-sized sectors.

Two of them are win and one of them is lose.

Now with C, it kinda looks like the same proportion of the spinner is taken with win, but it we don't know for certain whether it's 2/3 and actually if you look really carefully, you can see actually there's a little bit less of that spinner is taking up a win than would be there, so that's not 2/3, even though it might look close to it.

Here we have another spinner.

What is the probability that this spinner lands on win each time it's spun? Pause the video while you write down an answer and press play when you're ready to continue.

The answer is 3/7.

That's because there are seven equally sized sectors in that spinner and three of them say win.

Over to you now for task A.

This task contains two questions and here is question one.

Here we have four win-lose spinners, and you've got three questions you need to answer about those spinners.

So pause a video while you have a go at these questions and then press play when you're ready for question two.

And here is question two.

We have a spinner that is split into sectors which are all of equal size and each sector has a name in it, Alex, Jun or Sam.

Alex, Jun and Sam spin the spinner 40 times in total.

They get a point each time the spinner lands on their name.

Now, based on the scenario, you have some questions to answer.

Pause the video while you have a go at these and press play when you're ready to go through some answers.

Right, let's now go through some answers to question one.

So we have our four win-lose spinners.

Part A says, for each spinner, find the probability that it lands on win in a single spin.

So let's take a look at those.

The probability it lands on win for A is 2/3, for B is 5/8, for C is 2/5 and for D is 3/4.

Or you can express these percentages using any other equivalent fractions, decimals or percentages.

Part B says, Sophia wants to choose the spinner that gives her the best chance of winning.

Which spinner should she choose? So we wanna choose a spinner that has the greatest probability of win.

So out of those four probabilities we've written there, which one is the greatest? Well, that would be spinner D, 3/4 is the greatest out of those four fractions.

So that means spinner D has the best chance of getting a win.

Which spinner is the least likely to land on win? So out of those four probabilities, which is the least? It's spinner C.

That's because 2/5 is less than the values of those other three fractions.

And then question two, Alex, Jun and Sam spin the spinner 40 times and they get a point each time the spinner lands on their name.

Part A says, who should expect to get the most points? Well, that would be Sam, because Sam's name is written on that spinner more than anyone else's name.

And part B, who should expect to get the fewest points? That'll be Alex 'cause Alex's name is written the fewest number of times compared to all other people on that spinner.

And as part C says, what is the probability that the spinner lands on Alex? Well, there are eight equally-sized sectors on that spinner.

Only one of them is labelled Alex.

So the probability would be 1/8.

What is the probability the spinner lands on Jun? Well, once again, there are eight equally-sized sectors on that spinner, and three of them are labelled Jun.

So that means it's 3/8 for the probability it lands on Jun.

And what is the probability the spinner lands on Sam? Well, there are eight equally-sized sectors and four of them are labelled Sam.

So you can have 4/8 or you might have any other equivalent fractions, decimals or percentages to these.

And then part F says, how many points should Sam expect to get compared to Alex? So we have a bit of work to do here.

We need to first think about how many points Sam should expect to get and how many points Alex should expect to get, and then find the difference between those.

Sam would expect to get 20 points because that is 4/8 of 40 or half of 40 is 20.

Alex would expect to get five points.

That's 1/8 of 40.

So the difference between those is 20 subtract five, which is 15.

And part G, will your answer to F necessarily happen when they spin the spinner? Explain your answer.

Because what we put for part F there, that's what we expect to happen theoretically based on the probabilities.

But in reality, no, these results are only theoretical.

The actual results may differ just due to chance.

Some might get 20 points, some might get more than 20 or less than 20.

It's just the expected amount is 20 based on what we can see on this spinner.

Great work so far for the spinners.

Now let's apply what we've learned to some different events.

Here we have Jacob and Izzy.

They are finding the probability of rolling the number five on a fair six-sided dice.

Let's hear what they both think.

Jacob says, "I think the probability of rolling a five is 5/6." Izzy says, "I think the probability of rolling a five is 1/6." Who do you think is correct out of these two people? Pause the video while you have a think about this and press play when you're ready to continue.

Well, the probability that the dice lands on the number five is 1/6 and Izzy's gonna explain why.

She says, "The dice is equally likely to land on each of the six faces, and those six faces have one, two, three, four, five and six labelled on them.

That means the number five is written on one out of the six faces.

So the probability is 1/6, one for the number of fives and six for the number of total possible outcomes." So Jacob says, "So the size of each number on the dice doesn't affect the probability that it's rolled." That makes a lot of sense.

You're not more likely to get a five and a four, those are both equally likely to happen.

So it wouldn't make sense if the probability for five was greater than the probability for four.

All of them are 1/6.

Here we have a different scenario.

A bag contains blue cubes and green cubes.

All the cubes are of equal size.

Aisha closes her eyes and picks a cube out of the bag.

What is the probability that it is green? Maybe pause the video while you think about this yourself and then press play when you're ready to continue.

Well, the probability that it's green is 3/8.

The three comes from the number of green cubes in the bag, and the eight comes from the total number of cubes in the bag altogether.

And remember, all those cubes are all equally likely to be chosen from the bag.

So the probability is 3/8.

Here we have a different bag which still contains eight cubes where three of them are green and five of them are blue, but they look a bit different this time.

Can you see how they look a bit different to last time? The difference is the fact that the green cubes are bigger than the blue cubes.

So Lucas and Aisha are discussing whether the probability of choosing green would still be 3/8 if the cubes were different sizes.

Lucas says, "I think the probability would still be 3/8 because there are three green cubes and eight cubes altogether in total." Aisha says, "I think it won't be because all the cubes are not equally likely to be chosen." Who do you agree with this time, Lucas or Aisha? Pause the video while you think about that and then press play when you're ready to continue.

We should remember that theoretical probabilities can be found in scenarios where a trial has equally likely outcomes.

So for example, with the bag on the left, yes, you are more likely to get a blue than a green cube, but when we look at each individual cube, you are equally likely to draw any of those eight cubes out.

So all cubes are equally likely to be chosen, which means the theoretical probabilities can be found.

But the bag on the right, those green cubes are bigger than the blue cubes and we don't know how the probabilities of that big cube compares to a smaller cube.

So all the cubes are not equally likely to be chosen and the theoretical probabilities cannot be found.

So let's check what we've learned there.

Here we have the letters of the word mathematics.

A letter is going to be chosen at random from that word.

So what is the probability that the chosen letter is a vowel? Pause the video while you write down your probability and then press play when you're ready for an answer.

The answer is 4/11.

That's because four of the letters in the word mathematics are vowels, and there are 11 letters altogether.

Here we have a menu with lasagna, ravioli, bolognese and pizza as your options.

True or false, the probability that somebody chooses a pizza from this menu is 1/4.

Is that true or is it false? And then, choose one of the justifications below.

Pause the video while you make your choices and then press play when you're ready for an answer.

The answer is false.

Even though there are four outcomes and one of them is pizza, those four options may not necessarily be equally likely to be chosen.

For example, it might be that one of those items is more popular than the others and that will affect the probabilities.

So the probability is not necessarily 1/4.

Over to you now for task B.

This task contains two questions and here is question one.

Here we have Andeep and Lucas.

They each have a deck of cards spelling out their own names.

They play a game where they each randomly select a card from their own deck, record the letter and then return the card to the deck.

They win a point if they draw a card containing the letter A.

So based on the scenario, you have three questions to consider.

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

And here is question two.

A regular six-sided dice is rolled.

You need to find the following probabilities to begin with and then answer some questions based on those probabilities.

And then, when you've completed those questions, if you do want to investigate parts (j) and (k) further, you can click on the link to open up a simulation of rolling a dice 60 times.

Pause the video while you have a go at this and then press play when you're ready to go through some answers.

Okay, let's see how we got on with that.

So with Andeep and Lucas's deck of cards, they win a point if they draw a card containing the letter A.

What are the probabilities for each player winning a point? Well, for Andeep, it is 1/6 because there is one card of A on it and there are six cards altogether.

And for Lucas, it's 1/5 because there is one card of A in and there are five cards altogether.

Part B says, who is the most likely to get a point when they select a card? They each only have one card of the letter A in, but Lucas is more likely than Andeep because 1/5 is greater than 1/6.

They change the rules so that they get a point if they select a vowel.

So now who is most likely to get a point with these rules? And explain your answer.

Well, this is Andeep.

This is because the probability that Andeep picks a vowel is 3/6, while the probability that Lucas picks a vowel is 2/5 and 3/6 is greater than 2/5.

Or we can explain that as the probability that Andeep picks a vowel is a half, while the probability that Lucas pick a vowel is less than a half.

And then question two, we need find the following probabilities when a regular six-sided dice is rolled.

The probability it lands on three is 1/6, the probability it lands on four is 1/6 as well.

Those are both equally likely.

The probability it lands on not four, well, there are five numbers that are not four, so it's 5/6.

The probability it lands on even is 3/6.

There are three even numbers on a dice.

The probability it lands an odd is 3/6.

Now, for each of those, you could have 1/2 or any equivalent fraction to it as well.

For F, the probability it lands on a multiple of three is 2/6.

Those multiples of three are three and six, and there are six numbers altogether.

The probability it lands in seven is zero.

It's impossible to roll a seven on a dice because there's no seven on there.

And then H, the probability of not rolling a seven is one.

That's because all of the outcomes on a dice are not seven, so you are certain not to roll a seven.

The probability of rolling a factor of 60 is one whole as well.

That's because one, two, three, four, five and six are all factors of 60.

So you are certain to roll a factor of 60.

Any other equivalent fractions, decimals or percentages can also be used for these probabilities as well.

J says the dice is roll 60 times.

Approximately how many times could it be expected to land on the number three based on its probability? Well, that would be 1/6 of 60, which is 10.

And then part (k) says, will your answer to part (j) necessarily happen? Explain your answer.

Well, we expect it theoretically to land on number three approximately 10 times, but that's not necessarily what will happen.

For example, if you run a simulation of rolling a dice 60 times, here are some possible results you might get.

With the bar chart on the left, what we can see is we don't get three very much.

We get maybe six threes, but we get loads of ones, we get 20 ones.

Whereas with the example on the right, in that simulation, we rolled a three it looks like about 15 or 16 times.

That's definitely more than what we expected.

So even though the theoretical probabilities tell us what to expect, that might not necessarily be what happens, especially when there's only 60 rolls on the dice as well.

Fantastic work today.

Let's now summarise what we've learned.

A theoretical probability is a probability based on counting outcomes in scenarios where all individual outcomes are equally likely to happen.

The probability of an outcome or a set of outcomes can be found by considering the list of all the possible outcomes.

And a list of all possible outcomes can help us to find a probability even when the sets of outcomes are not equally likely.

For example, if you've got a spinner where three of them say A and two of them say B, we can still get a theoretical probability from that so long as each sector is equal sized or each individual A is equally likely to get.

Great job today, thank you very much.