Lesson video

Hello, you made a great choice for this lesson today.

It's gonna be a good one.

My name is Dr.

Olson and I'll be guiding you through it.

So let's get started.

Welcome to today's lesson from the unit of conditional probability.

This lesson is called Conditional Probability in a Two-Way Table and by the end of today's lesson we'll be able to calculate a conditional probability from a two-way table.

The lesson will introduce two new keywords.

One of them is conditional probability, which is the probability of an outcome occurring given that another event has occurred.

The other key word is independent.

Event A is independent of event B if the probability of event A occurring is not affected by whether or not event B occurs.

We'll see some examples of these during the lesson.

Here are also some previous keywords that will be useful during the lesson.

You may want to pause video if you need to remind yourself what any of them mean and press play when you're ready to continue.

The lesson is broken into two learning cycles and we're going to start with conditional probability on a two-way table.

Two-way tables are a great way of separating a sample space into groups based on an event, such as a characteristic of the population.

For example, here we have a box that contains four types of marbles.

Some of them are blue and contain a swirl.

Some of them are blue but don't have a swirl.

Some are green with a swirl and some are green without a swirl.

We can represent the frequency of each type of marble in a two-way table, such as the one you can see on a screen.

We can see the frequency of each type of marble, where the columns shows how many contain swirls and don't contain swirls, and the row shows how many are blue and how many are green.

So let's check what we've learned.

Here are five questions based on this two-way table.

Pause the video while you write the answers down to these questions and press play for some answers.

Let's go through some answers.

How many marbles in total are in the box? That'll be 60.

We can see that on the bottom right corner of the two-way table.

How many blue marbles are in the box? That'll be 29.

We can see that as the total of the blue row.

And then a random marble is chosen from the box and you have to find some probabilities.

For C, you have to find a probability that is blue with a swirl and that would be 18/60.

That is 18 because there are 18 marbles which are blue with a swirl and 60 because there are 60 marbles in total in the box.

The probability that the marble is blue would be 29/60.

The 29 comes from the total number of blue marbles and the 60 comes from the total number of marbles in the box and then the probability of drawing a marble that is not a green one with a swirl would be 45/60.

Now we can't see the number 45 in that table, but we can get it in a couple of different ways.

One way could be to add together the frequencies of all the marbles that are not green swirled marbles.

That would be by doing the blue marbles plus the green ones which don't have a swirl.

So 18 plus 11 plus 16 would give 45.

Another way could be to subtract the number of green swirled marbles from the total number of marbles.

That'll be 60, subtract 15 and that will give 45.

And the denominator is 60 because we are looking for the probability of drawing one from the box.

The probability of a certain marble being selected can change when certain conditions are met.

For example, a marble is randomly chosen, a part of the marble is now visible, and we can tell for certain that it's blue.

However, we cannot yet tell whether the marble has a swirl or not.

So based on the fact that we already know the marble is blue, what is the probability that it is a marble that contains a swirl? Pause the video while you think about how we might get this and press play when you're ready to continue.

Let's take a look together.

Since we already know that it's blue, it is guaranteed to be one of these 29 blue marbles, so we don't need to look at the entire two-way frequency table.

You can just look at this part here for blue marbles.

And then think to ourselves, "Well, 18 out of these 29 marbles have a swirl." So the probability of drawing a swirled marble in this case is not out to be outta 60, it's going to be outta 29.

It'll be 18/29 where 29 is a number of blue marbles and 18 is a number of those that contain a swirl.

Now this is something called conditional probability.

When we didn't know what colour the marble was and we're just drawing a marble from the box, the probability of drawing a blue swirled marble was outta 60 because it could have been any of those 60 marbles, but since we knew it was blue, the probability of it being a blue swirl marble changed because we had information already that it was definitely blue, so we knew it couldn't be any of the 60 marbles.

It had to be one of the 29 marbles and that affected the probability.

And that's why it's called conditional probability.

So in this case, the probability that it has a swirl, given that it is already blue, is 18/29.

The word given in conditional probability can be written with its own symbol and that is with a line symbol that you can see there, a vertical line.

If you see that vertical line in a bit of notation, that is about probability, it means the word given.

Here's an example.

This says P, and in the brackets it's got swirl.

It's got the vertical line and blue.

And what it means is the probability that it is a swirled marble given that you know already it's a blue marble, and that is 18/29.

The conditional probability of this being a blue swirl marble is different to a probability if we didn't know anything about the marble at all.

Let's compare these two probabilities and summarise what we've learned here.

The probability of drawing a marble that is swirl and blue is 18/60, whereas the probability of drawing a swirl marble, given that you already know it's a blue marble, is 18/29.

The difference between the notation here is the word "and" in given, which you could write as an intersection symbol and a vertical line if you want to, but also the difference in the values is in the denominator.

Both cases the numerator is the same, but the denominators differ because what you are drawing out of differs.

With the top one it implies that you are drawing, A, the 60 marbles, and what you're looking for are the ones that are blue and swirled.

But in the bottom probability the word given implies that you already know it's blue, so you're not selecting from any of the 60, you know you're selecting from one of the 29 and that's why that denominator is different.

So let's check what we've learned.

A different box also contains blue and green marbles, each of which can either have a swirl or not have a swirl.

A random marble is selected.

Please could you find the probability that it's green and has a swirl? Pause the video while you write down your answer and press play for an answer.

The answer is 25 over 140.

The wording of this probability doesn't suggest that we know anything about the marble already.

We don't know whether it has a swirl.

We don't know whether it's green.

We just know that we are drawing from one of the 140 marbles and 25 of them are green and swirl.

So the top of the marble we can see now is green.

That case, could you please find the probability that there is a swirl marble given that you now already know it's a green marble? Pause video while you write down your answer and press play when you're ready to see what it is.

Well, now we know it's green.

We know it can't be any of the 140 marbles.

We know it must be a marble from this row in the table.

There are 72 marbles which are green and we've selected one of them and we want the probability it has a swirl, so that would be 25 over 72.

Here's a different scenario.

Year 11 language pupils at both Oakville Academy and Elmsleigh School learn exactly one of either French, Spanish or German.

And the frequency two-way table shows you how many study each at each school.

One language pupil is chosen at random.

What is the probability that the pupil learn Spanish at Elmsleigh? Pause video while write down your answer and press play when you're ready to see what it is.

Well, there's nothing in the wording of this question that implies we know anything about this pupil already.

We don't know what school they attend.

We don't know what language they study.

So we are selecting from any of the 240 pupils.

That means their probability would be 54 over 240.

One language pupil is chosen random and they say that they learn German.

How many possible pupils could this be? Pause the video while you write it down and press play when you're ready to see the answer.

There are 110 pupils altogether who study German.

Some go to Oakfield Academy and some go to Elmsleigh School, but there are 110 altogether.

Given that this pupil learns German, what is the probability that they go to Oakfield Academy? Pause the video while you write down your answer and press play when you're ready to see what it is.

The wording of this question tells us that we already know something about the pupil.

We know that they study German.

So we are not selecting from any of the 240 pupils.

We are only selecting from the pupils who study German.

That means the denominator will be 110 and the numerator will be 86.

So it's 86 over 110.

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

This task contains three questions and here is question one.

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

And here is question two.

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

And here is question three.

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 for more answers.

Here are the answers to question two.

Pause while you check these and press play for more answers.

And here are the answers to question three.

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

Well done so far.

Let's move on now to mutual exclusivity and independence.

Here we have our box of marbles again, and Jacob says, "If I know some information about the marble, then the probability of a marble having a certain colour or pattern will always be different compared to if I didn't know that information at all." Sofia says, "I don't think that's true." She says, "The probability of a marble having a swirl doesn't change even if we know it's blue or green." What do you think about this? Who do you think is correct here between Jacob and Sofia? Pause the video while you think about this and press play when you're ready to continue.

Well, we can test Sofia's observation by comparing three probabilities about choosing a swirl marble from this box, and for this investigation we may need to simplify our probability fractions along the way.

One probability will be this one.

A random marble is chosen from the box.

We don't know anything about it at all.

What is the probability it has a swirl? That would be 10/60 because to our 10 marbles contain a swirl altogether out of 60 marbles and that would simplify to 1/6.

Here's another probability.

A random marble is chosen from the box and we can see already that it's blue.

So what is the probability it has a swirl given that we already know it's blue? Well, that would be a marble from this top row here.

There are 12 blue marbles.

We can see that two of them have swirls.

So the probability would be 2/12, which is 1/6.

And let's look at a third probability now.

A random marble is chosen from the box and we can see that it's green.

What is a probability that marble has a swirl given that we know it's green? We can find this probability by looking at the second row in this table.

We can see there are 48 green marbles altogether in the box and eight of them have a swirl.

So the probability would be 8 over 48, which also simplifies to 1/6.

So once again, the probability it has a swirl given it's green is the same as a probability it has a swirl just generally from the box.

So now we've seen three cases where the probability of a marble having a swirl is the same each time, whether it's drawn from the entire box, whether it has a swirl given it's blue, or whether it has a swirl given it's green.

All those probabilities are 1/6.

So even if we are given information about the marble, the probability that the marble is a swirl marble does not change.

The probability of the marble having a swirl is 1/6 every time, and this is because the probability it has a swirl is independent of its colour.

An event or outcome is independent of a property if the probability of it occur does not change when given information about that property.

Let's look at this word independent in a little bit more detail now.

An event or outcome is independent if it satisfies either pair of conditional probabilities that we're about to look at.

One pair is this, that the probability of the outcome, given that a property is met, is equal to the probability of the outcome given that the property is not met.

If those two probabilities are equal, it means the outcome is independent of the property because regardless of whether that property is met, the probability of the outcome happening is the same.

The other pair of conditional probabilities is this one here, that the probability of the outcome happening given that the property is met, is equal simply to the probability of that outcome happening in general.

If those two probabilities are the same it means the probability of the outcome happens is the same regardless of whether or not that property is met, so the outcome is independent of the property.

Let's take a look an example of this now by considering the two-way frequency table we can see on the screen.

For the trial of randomly choosing a marble from the box, the outcome of choosing a swirl marble is independent of it being the colour blue because, and we can justify it in two ways.

One way is because the probability it has a swirl, given that we know it's blue, is equal to the probability it has a swirl given that it's not blue.

The probability has a swirl given that it's blue is 2/12 and the probability it has a swirl given that it's not blue is 8/48.

And those two fractions simplify to the same thing.

They both simplify to 1/6 Another way we can justify it is by saying the probability it has a swirl given that we know it's blue is equal the probability it has a swirl in general.

The probability has a swirl given that it's blue is 2/12 and the probability has a swirl in general is 10/60.

Both of those fractions simplify to the same thing.

In every case each probability is exactly 1/6.

So the fact that each of those pairs of probabilities are equal to each other, it means whether it has a swirl is independent or whether it's blue.

So let's check what we've learned.

Year 10 language pupils at Oakfield Academy and Elmsleigh School learn exactly one of either French, Spanish or English.

And the two-way frequency table shows us how many pupils at each school study each language.

One language pupil is chosen at random.

What is the probability that the pupil goes to Oakfield Academy? Pause the video while you write it down and press play for an answer.

The answer is 100 over 250, which simplifies to 2/5.

One language pupil who learns Spanish is chosen at random.

What is the probability that the pupil goes to Oakfield? Pause the video while you write down your answer and press play when you're ready to see what it is.

The answer is 20 over 50, which also simplifies to 2/5.

And here's one more.

One language pupil who doesn't learn Spanish is chosen at random.

What is the probability that the pupil goes to Oakfield? Pause the video while you write down your answer and press play when you're ready to see what it is.

The answer is 80 over 200, which also simplifies to 2/5.

Which of these statements is correct based on what you've just done? Pause video while you choose and press play when you're ready for an answer.

The answer is B.

The probability of a student going to Oakfield is independent of whether they learn Spanish or not.

And we can explain that with either these two justifications.

The probability go to Oakfield given that they learn Spanish is equal to the probability that they go to Oakfield given that they do not learn Spanish.

Or you can justify it by saying the probability they go to Oakfield given that they learn Spanish is just equal to the probability to go to Oakfield in general.

Let's now look at another scenario involving marbles in a box, which is similar to the previous one but just a little bit different.

A different set of 100 marbles can either be blue or green, only this time they can also have either a swirl, a sparkle, both a swirl and a sparkle, or neither a swirl nor a sparkle.

And the two-way frequency table shows us some information about how many of each marble there is in the box.

However, Sofia says, "This two-way table is pretty unclear since I don't know how many marbles have both a swirl and a sparkle.

And the total doesn't make sense either.

There are exactly 100 marbles in the box but 14 plus 20 plus 13 plus 18 plus 22 plus 23 gives 110." Those six frequencies sum to 110, not 100.

Can you see why that might have happened? Pause the video while you think about this and press play to continue.

The reason the total in this table is different from the real number of marbles is because some marbles have been counted twice.

If a marble has both a swirl and a sparkle, then it is counted once in the swirl event and once in the sparkle event.

Therefore, 10 marbles have been counted twice, making the total of the table 10 higher than it should be.

This is because these events, and by these events I'm referring to where it has a swirl or a sparkle or not, these events are not mutually exclusive.

Mutually exclusive events cannot happen at the same time.

However, a randomly chosen marble can in this case have both a swirl and a sparkle at the same time.

So the fact that both events can happen at the same time means that they are not mutually exclusive.

If you look at blue and green instead, a marble cannot be blue and green.

So blue and green are mutually exclusive.

But where it has a swirl and sparkle, they are not mutually exclusive 'cause they can happen at the same time.

To fix this in the table, we can create a separate event specifically for marbles that have both a swirl and a sparkle.

These four events are now all mutually exclusive from each other.

So we have some which are only have a swirl, some which have only have a sparkle, some which have both, and some much have neither.

Now we separate them out.

They are mutually exclusive events.

Because we can't have a marble that only has a swirl and only has a sparkle.

Those two things don't make sense to happen at the same time.

So now the events are mutually exclusive.

And we can see if we add together the frequencies, we do get 100 this time.

Here's another set of 100 marbles.

They can either be blue or pink, they can have a swirl or not have a swirl, they can have a sparkle or it can have both a swirl and a sparkle.

And we can see the frequencies for these in a two-way frequency table.

For a two-way table to work, we need mutually exclusive and exhaustive events on both the row and the column headings.

However, it is possible to find other mutually exclusive events on a table.

How might you spot these on this two-way table? Pause the video while you think about this and press play when you're ready to continue.

Mutually exclusive means events that cannot happen at the same time.

We know we can't get a blue neither marble because there is a frequency of zero in the table.

Therefore the events blue and neither are mutually exclusive.

So let's check what we've learned.

There are 120 students who learn languages in Birchwood Academy and there are 130 students who learn languages at Willowsford School.

And the two-way frequency table shows you how many of each student studies each language in each school.

Now let's just think about the choice of language for a second.

If choosing French and choosing Spanish and choosing German are all mutually exclusive to each other, it means a pupil cannot choose more than one them at the same time.

So at which school are the languages of French, Spanish and German not mutually exclusive? And how do you know? Pause the video while you work this out and press play when you're ready for answers.

Well, the way we can determine this is by adding up how many students study French, Spanish, and German in each school and comparing it to the total number of students in the school who study languages in general.

If we do that, we'd see these figures.

For Willowsford School if we add together the number of pupils who study French, Spanish and German, we get 130.

And that is the same as a number of pupils in total who study languages at that school, which means each person must study one language.

But for Birchwood, we can see that 28 plus 52 plus 57 gives 137, and that is more than the number of pupils altogether who study languages, which means there must be some pupils who study more than one language at that school.

Therefore, the school at which the languages French, Spanish, and German are not mutually exclusive is Birchwood Academy.

Therefore, this two-way table would need to be redesigned.

Here's another two-way frequency table.

You are told that events A, B, and C are mutually exclusive and that events D, E, and F are mutually exclusive as well.

Could you please look at the frequency table and write down two additional pairs of mutually exclusive events that you can find in this table? Pause the video while you do that and press play when you're ready for answers.

You are looking for events that do not happen at the same time and they are A and D and C and E.

And this is because the probability of A and D happening is zero and the probability that C and E both happen is also zero.

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

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

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

And here are the questions for three and four.

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

Okay, let's go through some answers.

Here are the answers to question one.

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

Here are the answers to questions two A and B.

Pause while you check these and press play for the rest of question two.

Here's the answer to part C of question two.

Pause while you check this against your own and press play for more answers.

Here are the answers to question three.

Pause while you check these and press play for more answers.

And here are the answers to question four.

Pause while you check these against your own and press play for today's summary.

Fantastic work today.

Now let's summarise what we've learned.

The conditional probability of an event occurring is a probability of the event occurring given that a particular condition is met or known, such as, for example, one property of an object already being known.

The conditional probability of an event given a particular condition is met may be different to the probability of that same event without any conditions being known.

If the two probabilities are the same, then the probability of this event occurring is independent of that condition being known or not.

And two mutually exclusive events cannot happen at the same time, and so the combined probability is always zero.

Well done today.

Have a great day.