Lesson video

Hello there, it's a great day for a great lesson.

My name is Dr.

Rowlandson and I'll be guiding you through it.

Let's get started.

Welcome to today's lesson from the unit of Conditional Probability.

This lesson is called Checking and Securing Calculating Probabilities from Tables.

And by the end of today's lesson, we'll be able to calculate probabilities from tables and two-way tables.

Here are some previous keywords that'll be useful during today's lesson.

You may want to pause the video if you need to remind yourself what either of these words mean, and then press play when you're ready to continue.

The lesson is broken into two learning cycles.

In the first part of the lesson, we're going to be looking at tables that display outcomes.

And in the second part of the lesson, we're going to be looking at tables that display the frequencies of outcomes.

And in both parts, we're going to be using those tables to find probabilities.

Let's start with probability and tables of outcomes.

Here we have a spinner that contains some numbers.

Jacob says, "I can split the outcomes of this spinner into a pair of events: square numbers and non-square numbers.

And represent these on a table." Something a bit like this.

The left-hand column of this table shows all the outcomes for the spinner that satisfy the event of it being a square number.

The right-hand column of the table shows all the outcomes that satisfy the event of it being a non-square number.

Now here's Sofia, who says, "What if I also wanted to group the outcomes into a second pair of events: odd and even numbers?" Well, Jacob says, "We can split a table up and move the outcomes." A bit like this.

Now each cell in this table shows the outcomes which satisfy two events.

For example, the cell in the top left corner of this table has the numbers 1 and 9 in.

These are the outcomes that satisfy both the events of it being a square number and an odd number.

And the same with the other cells in the table as well for their events.

We can find the probability of an outcome by counting its frequency on the table.

For example, if we want to find the probability of the spinner landing on 16, it would be 2/10.

The 2 is the number of 16s, which we can see on the spinner, and we can also see that in the table as well.

10 is the total number of equally likely outcomes.

We can see that on the spinner by counting the sectors, and we can see it in the table by counting the numbers in the table.

In a similar way, we can find the probability of an event by using the frequency of outcomes that satisfy the event.

For example, if we want to find the probability of this event here, that the spinner lands on an even square number, then we could look at this cell in the table here.

This cell represents all the outcomes which are square and even numbers, and we can see that there are three numbers in that cell.

So the probability will be 3/10.

3 is the frequency of outcomes that satisfy this event and 10 is the amount of outcomes altogether.

Here's another example.

We could find the probability of getting a non-square number with this spinner.

We can do that by looking at two cells in this case, or an entire column, that's the column for non-square numbers.

We can see that the frequency of outcomes in this event is 5 because there are 5 numbers that are currently highlighted in that table.

The number 5, 7, 11, 15, and 20.

So the probability would be 5/10 because there are 5 outcomes that satisfy the event out of 10 possible equally likely outcomes.

So let's check what we've learned.

Here we have some drinks which are labelled from A to I.

These drinks need to be sorted into the outcome table according to whether they have a handle or a straw.

You can see that there's one cell on this table highlighted.

Which drinks would go in this cell? Write down the letters, pause the video while you do that, and press play when you're ready for the answer.

The answer is B, C, F, and I.

Now the outcome table is fully completed.

So the drinks have been sorted according to whether they have a handle or a straw.

A drink is chosen at random.

Find the following three probabilities.

A is the probability that it has a straw with no handle, B is the probability that it has no straw, and C is the probability that it either has a handle or it has a straw, but not both.

Pause the video while you write down those three probabilities and press play when you're ready for the answer.

Let's go through some answers.

A, the probability of choosing a drink which has a straw with no handle would be 3/9.

B, the probability of it having no straw would be 2/9.

And C, the probability of it either has a handle or a straw, but not both, would be the frequency from these two cells, which would be 4/9.

A table of outcomes can also show the outcomes of a two-stage trial.

For example, a fair 4-sided dice with numbers 1, 2, 3, and 4 is rolled twice.

So those are our two stages, rolling it once and rolling it again, and the outcomes of each roll are added together.

So, here we have an outcome table, where the outcomes of each roll are the headers of the table.

And the combined outcomes, which we get from adding the two outcomes together, is written in each of the cells.

And we can see that here.

For example, in the bottom right corner, you can see the number 8.

That is the combined outcome of roll one and roll two, 'cause you get 4 and 4, which add together to get 8.

In order to calculate the probability of an outcome or an event, we need to count the frequency of relevant outcomes across all parts of the table.

For example, if we want to find the probability that the outcome from this two-stage trial is 5, or, in other words, that the two rolls of the dice sum to 5, then we need to look at how many times 5 appears in the outcomes on this table and how many outcomes there are altogether.

The frequency of 5 in this table is 4.

We can see that there are 4 cells with the outcome 5 written in.

So the numerator of our fraction for the probability will be 4, and there are 16 outcomes altogether, so the probability will be 4/16, or anything equivalent.

Another example would be this event here, the probability that we get a prime number that is under 5.

Well, we can look at how many of these outcomes are prime numbers that are under 5.

Here are all the prime numbers that are under 5, so the probability would be 3/16.

So let's check what we've learned.

A trial consists of the following two stages: Stage one is flipping a fair coin, with heads or tails as the possible outcomes.

Stage two is rolling a fair six-sided dice, with the integers 1 to 6 as the possible outcomes.

Now, there's some rules to this.

If the coin lands on heads, the result of the dice is doubled.

If the coin lands on tails, the result of the dice remains the same.

Here we have an outcome table to show the combined outcomes of this trial.

You can see that some examples are given to you.

If the coin lands on heads and the dice roll is 5, then the outcome is 10 because 5 is doubled.

If the coin lands on tails and the dice roll is 5, then the final outcome is 5 because the dice roll remains the same.

You can see that there are three cells with A, B and C in.

Could you please calculate the values of A, B, and C? Pause the video while you do that and press play when you're ready for an answer.

The answers are 2, 3, and 12.

The table is now fully completed.

Could you please use the table to calculate the following probabilities? In A, could you find the probability that the outcome is odd? And B, could you find the probability that the outcome is a single-digit even number? Pause the video while you do these and press play for answers.

Let's look at some answers.

For part A, the probability that the outcome is odd is 3/12.

We get the 3 because there are 3 outcomes in this table which are odd.

They are 1, 3, and 5.

And the frequency for each of those outcomes is 1, so altogether the frequency is 3 out of 12, that's why it's 3/12.

The probability of getting a single-digit even number would be 7/12.

That's because there are 7 cells in that table that contain a single-digit even number.

In some cases, we have the same number twice, for example, 4 appears twice in the table, but the total frequency for single-digit even numbers in that table is 7 out of a total frequency of 12 altogether, so it's 7/12.

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

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

Pause the video while you do these and press play when you're ready for more questions.

And here is question three.

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

And here is question four.

Pause while you do this and press play for some answers.

Okay, here are the answers to question one.

Pause while you check these against your own and press play when you're ready for more answers.

Here are the answers to question two.

Pause while you check this and press play for some more.

Here are the answers to question three.

Pause while you check and press play to continue.

And question four look like this, and here are the answers.

Pause while you check these and press play for the next part of the lesson.

Well done so far.

Now let's move on to the next part of this lesson, which looks at probability and two-way tables of frequencies.

Here we have an outcome table that shows all the outcomes from a two-stage trial.

That two-stage trial involved rolling a four-sided dice with numbers 1 to 4 on and rolling it again and then finding the sum of those two outcomes.

And that's what we can see in the table here.

Now, here we have Jacob, who's not concerned with all of this table.

He says, "I'm only concerned with two events: prime numbers and numbers less than 5.

Could I use a table that just shows the information that I want?" Well, rather than showing every single individual outcome for a trial, a table can be used to show the frequencies of combined events.

And it can look something a bit like the table on the right-hand side.

The columns of that table show an event and its complement, either getting under 5 or getting 5 and above.

The rows of the table show another event and its complement, getting prime and getting a number that is not prime.

Each cell on the table has exactly one value, and that is the frequency of outcomes that satisfies its pair of events.

Let's fill it in together.

The cell in the top left of that table should show the frequency of all the outcomes that are both under 5 and are prime numbers.

Those are highlighted in the outcome table and they are 2 and 3 and 3.

So the frequency is 3.

The cell on the bottom left of the frequency table shows the frequency of all outcomes that are under 5 and are not prime.

We can see that the frequency of those is 3 as well.

The cell in the top right of the frequency table shows the frequency of all outcomes that are 5 and above and are prime.

Well, by looking at the outcome table on the left, we can see that frequency is 6.

And the final cell in the bottom right of this frequency table should show the frequency of all outcomes that are 5 and above and are not prime.

And we can see on the outcome table there are 4 outcomes that satisfy that event, so the frequency would be 4.

And then when it comes to finding the probability of an event, such as the event that the outcome is prime and under 5, we could use either of these tables to do that.

If we want to use the outcome table on the left, we would have to count all the individual outcomes that satisfy this event.

If we want to use the frequency table on the right, that information is already given to us.

The top left cell of that frequency table says there are three outcomes which are under 5 and prime numbers.

So, we can get the probability as 3/16.

The 16, which is in the denominator of that fraction, that could also be taken from either of the tables.

Using the left table, the outcome table, you'd have to count all the outcomes and count that there are 16 cells altogether.

In the frequency table, you'd get the 16 by adding together all of the frequencies, 3 + 3 + 6 + 4, and that would give 16 as well.

So let's check what we've learned.

A trial is conducted.

The outcomes of this trial are shown in the outcome table in the bottom left of the screen.

You can see also there's a frequency table in the bottom right of the screen, but there are unknowns in that frequency at the moment and they are labelled A to D.

Could you please use the outcome table to find the values of A, B, C, and D in the frequency table for the same trial? Pause the video while you do that and press play when you're ready for answers.

Okay, here are the answers.

A is 4 because there are 4 outcomes in the outcome table which are under 6 and are even.

They are 2, 2, 4, and 4.

Yes, there are only 2 different outcomes, but each of those different outcomes has a frequency of 2, so the total frequency altogether is 4.

And you can apply a similar method to get B as five, C as 3, and D as 0.

So, could you now use this frequency table to find the probabilities displayed on the screen? One of them is not possible based on this information.

See if you can spot which one it is.

But the rest are.

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

Okay, let's look at some answers.

If we write our probabilities as fractions, then the denominator will be the total frequency, which we can get from this table by doing 4 + 5 + 3 + 0.

So the denominator will be 12.

So, that means we need to look now for what the numerator would be in each case.

For part A, the event that it's an even number under 6 is shown in our frequency table in the cell that is under 6 and even, and the frequency in that cell is 4.

So the probability would be 4/12.

For part B, the event that the numbers are 6 and over is represented in two of those cells, that is, 5 and 0, so the probability will be 5/12.

For part C, the event that the number is 6 and over and is not even, that is represented in one cell and it would be 0/12, that is impossible to happen.

And then for part D, the probability that it's a square number, well, this frequency table doesn't tell us how many of the outcomes are square numbers, so that's not possible from this frequency table.

When conducting an investigation, we may collect data on a large population.

We can break down this population into groups, based on a characteristic of the population, and that's called strata.

We can break a population down into two sets of groups and to display the amount of members for each group by using a two-way frequency table.

Let's take a look at an example.

Here we have a two-way frequency table, which we're going to fill in shortly.

But before we look at the context for this table or any of its information, I'm going to adapt it slightly.

When displaying large sets of data on a frequency table, it can help to add an extra column and an extra row to show the total frequency of each group, like this.

It is possible to construct and complete a frequency table if we are given enough information about the population.

So let's get some more information about the context and some of the information about this population.

A survey of 400 people were asked whether they lived in Oakfield or Rowanwood, and whether they shopped more online or in-store.

So already, based on this information, we can fill one cell in on this table, and that would be the 400.

The 400 is the total population, so you can put that on the bottom right corner of this table.

Let's get some more information.

250 of the people surveyed live in Oakfield.

We can put that in our table as well.

It would go here.

It is the total number of people who live in Oakfield that was surveyed in this table.

Now we've got this, we can calculate some other missing information.

We can use the grand total of 400 to figure out how many people live in Rowanwood, and that'll be by doing 400 subtract the 250 people from Oakfield to get 150 people in Rowanwood.

Here's some more information about that survey.

160 people from Oakfield shop in-store.

So we can fill that in on the cell that is for Oakfield and in-store.

And we can use that to find some other missing numbers.

We can use the Oakfield total of 250 to figure out how many people live in Oakfield shop using online delivery.

And that would be by doing 250 - 160, which is 90.

We are also told that a total of 140 people shop online, and that is now displayed in our table for the total of online shoppers.

We now have enough information on the table to calculate two more frequencies, and that is the number of people in Rowanwood in the survey who shop online and the total number of people in the survey who shop in-store.

Can you think how we might get these frequencies? Pause the video while you consider this and press play when you're ready to continue together.

Well, we can get the number of people in Rowanwood who shop online by taking the total number of people who shop online in this survey and subtracting the number of those who live in Oakfield.

That'll be 140 - 90, which is 50.

We can calculate the number of people who shop in-store altogether by taking the total number of people in this survey and subtracting the number of people who shop online.

And that'll be 400 - 140, which is 260.

This final frequency can be calculated in multiple different ways, but let's focus on a particular, a horizontal calculation and a vertical calculation.

If we perform both these calculations and get the same answer each time, we can be quite confident that the rest of our calculations have been correct along the way.

For example, we could take the total number of people in Rowanwood in this survey and subtract the number of those who shop online to get the number of them who are shopping in-store.

And that would be 150 - 50, which is 100.

Another way could be to look at the total number of people who shop in-store in this survey, subtracting the number of those who live in Oakfield, and getting the number of people in Rowanwood who shop in-store that way.

We'd get 260 - 160, and, again, we get 100.

So the fact we've got the same answer in two different ways there makes us quite confident that the answer is correct.

100.

Now this frequency table is complete.

We can use it to calculate some probabilities.

For example, a random person from the survey is chosen.

The probability that the person is from Oakfield can be calculated by looking at the total number of people in this survey who are from Oakfield and writing that as the numerator of our fraction, and the denominator will be the total number of people in the survey altogether.

So, it'll be 250/400.

To get the probability that the person is from Rowanwood and shops online, we'd need to look at this cell here.

There are 50 people in this survey who are from Rowanwood and shop online, out of 400 people altogether.

So it would be 50/400.

We can also use this frequency table to calculate proportions from subgroups of the population.

For example, what fraction of the people in Oakfield shop in-store? So we want to know not the fraction of the overall survey, but the fraction of just the people in Oakfield.

That means the denominator of our fraction would be 250, and of those, 160 of them shop in-store, so our fraction would be 160/250, which simplifies to 16/25.

What fraction of people who shop in-store live in Oakfield? So this time we're looking for the fraction of people who shop in-store.

So our denominator is going to be the number of people who shop in-store altogether, which is 260.

And our numerator will be how many of those live in Oakfield, which is 160.

So our fraction would be 160/260, which is 8/13.

Let's check what we've learned.

Here we have a two-way frequency table which is incomplete.

It shows the number of students in class A and class B who either walk to school or use other transports.

Could you please find the values of the unknowns labelled A to E and find them in alphabetical order? Pause the video while you do that and press play when you're ready for answers.

Let's take a look at some answers.

A will be 41, which you can get from doing 86 - 45.

B will be 30, which you can get from doing 41 - 11.

C will be 25, which you can get from doing 45 - 20, or you could do 86 - 11 - 20 - 30.

But probably quicker the first way.

D would be 55, which you can get from doing 30 + 25.

And E would be 31, and there are lots of different ways you can get that.

You could do 11 + 20 or you can do 86 - 55.

Here are three questions to answer based on this frequency table now.

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

Okay, let's go through some answers.

For part A, one student is chosen at random from these 86 students.

You have to find the probability that they are from class A and walk.

That'll be 11/86.

The 11 comes from the fact that there are 11 people in that table who walk and are from class A.

B, what fraction of the students in class B use transport? Well, in class B, there are 45 students altogether and 25 of them use transport.

So the fraction of B, 25/45, which is 5/9.

And then in part C, you have to work out what fraction of the students who use transport are in class B.

So the wording of that question is just ever so slightly switched around, but that will affect how you get your answer.

'Cause this time, our denominator is gonna be the total number of people who use transport, which is 55, and the numerator will be how many of those are in class B, which is 25.

So it'll be 25/55, which is 5/11.

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

This task contains three questions and here is question one.

Pause the video while you do it and press play for question two.

Here is question two.

Pause the video while you do this and press play for question three.

And here is question three.

Pause the video while you do this and press play for answers.

Okay, here are answers to question one.

Pause while you check these against your own and press play when you are ready for question two.

And here are answers to question two.

Pause while you check these against your own and press play for answers to question three.

For question three, your two-way table should look something like this.

Pause while you check this against your own and press play to continue with answers for question three.

And then based on your answers for part A, here are your answers to parts B to E.

Pause while you check these and press play for the summary of today's lesson.

Great work today.

Now let's summarise what we've learned.

The outcomes from a trial can be shown on an outcome table and split into different events.

The outcomes from a two-stage trial can be shown on an outcome table, where the outcomes from each stage are combined in some way, such as numerical results added together.

The frequency with which different events on an outcome table occur can be represented on a frequency table.

The frequency tables can also be used to show a population that has been split up into different groups.

And probabilities can be calculated from different types of tables.

Well done today.

Have a great day.