Lesson video

Hello there, and thank you for joining me.

My name is Dr.

Ronson, and I'll be guiding you through this lesson.

So let's get started.

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

This lesson is called Checking and Securing Exhaustive Events.

And by the end of today's lesson, we'll be able to work probabilities as fractions, decimals and percentages and use the fact that probabilities of exhaustive events sum to one.

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

So you might want to pause the video if you want to remind yourself what they mean and press play when you're ready to continue.

The lesson is broken into two learning cycles.

In the first learning cycle, we're going to focus on what it means for events to be mutually exclusive and exhaustive.

And then the second learning cycle, we're going to be summing probabilities to one.

Let's start off with mutually exclusive and exhaustive events.

Here we have a regular six-sided dice, which has rolled once.

And we can see the sample space that shows all possible outcomes for this trial.

We get the numbers, 1, 2, 3, 4, 5 or 6.

Two events are considered.

Event A is the event that we roll an even number.

Event B is the event that we roll an odd number.

The outcomes that satisfy each of these events can be represented in a Venn diagram.

We have A, which is represented by one circle and B represented by another circle, and we can see that they overlap in this diagram here.

Let's take a look what happens when we fill in the outcomes on this Venn diagram.

The outcomes that satisfy event A, that the role is even are 2, 4 and 6.

None of those satisfy event B, so they are not in the overlap.

They are only in the section which is just for A.

The outcomes that satisfy event B, the ones which are odd are 1, 3 and 5.

Now, none of those satisfy event A, so none of those are in the intersection.

They're all in the section that only is for event B.

Now, if we look at this Venn diagram, you'll notice that there are two regions of it that are empty.

One is the intersection and one is the region outside the circles, at least tell us something about these two events.

These two events are mutually exclusive.

They are mutually exclusive because there are no numbers on a dice that are both odd and even.

It's impossible to roll a dice and get an outcome that satisfies both of these events, and that can be seen by the empty intersection on the Venn diagram.

These events are also exhaustive and that's because there are no numbers on a dice that satisfy neither of those events.

There are no numbers but are neither odd or even.

In other words, every single possible outcome for when you roll a dice is accounted for by either events A or B.

That's why the section on the outside the circles of Venn diagram is empty.

This Venn diagram could even be represented like this if we wanted to.

So let's use the same scenario but define two different events.

Event C is getting a factor of 10 and event D is getting a factor of 12.

Let's consider whether or not these events are mutually exclusive, whether they are exhaustive, and do that by concerning the Venn diagram.

So the outcomes on the Venn diagram would look something a little bit like this.

Now, we look at this Venn diagram, we'll notice that the region around the outside of the circles is empty.

That means these two events are exhaustive.

They're exhaustive because there are no numbers on a dice that fail to satisfy either event.

Every single outcome is somewhere in at least one of those circles, so they are exhaustive.

But if we look at the intersection, we can see that these two events are not mutually exclusive.

They're not mutually exclusive because there are some numbers on a dice that satisfy both events.

One is a factor of both 10 and 12, and so is 2.

The fact that there are in some numbers on a dice that satisfy both events means that they are not mutually exclusive.

Let's look at another pair events.

Event E is the event of rolling a multiple of 3 and event F is the event of rolling a multiple of 4.

Let's consider whether or not these events are mutually exclusive and whether or not they're exhaustive.

Pause the video while you think about this and press play when you're ready to continue together.

Let's take a look together and let's look by looking at the Venn diagram.

It would look something a bit like this.

Now, in this Venn diagram, we can see that the intersection is empty.

That's because these two events are mutually exclusive.

There are no outcomes that satisfy both events.

There are no numbers on a dice and that are both multiples of 3 and multiples of 4.

There are some numbers in general that are multiples of 3 and 4, such as 12, but 12 was not on a regular six-sided dice.

So in this case, those two events are mutually exclusive, but these two events are not exhaustive.

They are not exhaustive because there are outcomes but are not in subsets for either E or F.

In other words, these two events do not account for all possible outcomes when you roll a dice, and you can see that in the Venn diagram because there are some numbers that are not in the circles.

They're the numbers 1, 2 and 5.

So these are not exhaustive events.

Let's check what we've learned.

Here we've got a spinner, we've letters A to E.

And we've got two events that have been defined.

Event P is the event of the spinner landing on a consonant.

Event Q is the event that the spinner lands on a vowel.

True or false.

Events P and Q are mutually exclusive for a single spin of this spinner.

Is that true or false? Pause video while you choose and press play when you're ready for an answer.

The answer is true.

So why is it true? Write a sentence to explain why it's true and press play when you're ready for an answer.

It's true because the spinner cannot land on a letter that is both a consonant and a vow.

So true or false: events P and Q are exhaustive for a single spin.

Pause while you choose and press play for an answer.

The answer is true.

So why is that? Pause why you write it down and press play for an answer.

It's true because the spinner cannot land on a letter that is neither a constant nor a vowel.

So let's look at a different pair events.

P is now defined as getting an A, a C or an E, and Q is now defined as getting an A, a B or a D.

So true or false: events P and Q are mutually exclusive for a single spin of this spinner? Pause while you choose and press play for an answer.

The answer is false.

So why is that? Pause while you write something down and press play for an answer.

The answer is false because the letter A satisfies both events.

So if there's an outcome that satisfies both events, it means the events are not mutually exclusive.

So true or false: events P and Q are exhaustive for a single spin of this spinner.

Pause where you choose and press play when you're ready for an answer.

The answer is true.

Why is that? Pause while you write something down and press play for an answer.

It's true because there are no letters on the spinner that failed to satisfy either event.

All the letters in that spinner are somewhere in at least one of those two events.

So let's change the events again.

Event P is now either getting an A or a C.

Event Q now includes B and D.

So true or false: events P and Q are mutually exclusive for a single spin of this spinner? Pause while you choose and press play for an answer.

The answer is true.

So why is that? Write something down and press play when you're ready for an answer.

It's true because the spinner cannot land on a letter that satisfies both events.

And true or false: P and Q are exhaustive for a single spin.

Pause while you choose and press play for an answer.

The answer is false.

So why is that? Pause while you write something down and press play for an answer.

It's false because the letter E does not satisfy either event.

Not all of those letters on that spinner are accounted for by these two events, so it's not exhaustive.

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

This task contains three questions, and here is question one.

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

Here is question two.

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

And here is question three.

It is very similar to question two, but this time you're focusing on mutually exclusive.

Pause while you this and press play for some answers.

Okay, let's go through some answers.

For question one, you've got a bag of counters.

All the counters have letters A, B or C on, and they're either black or whites.

You have to write true or false for each statements.

Events D and E are mutually exclusive.

That would be true.

There are no counters that are both black and white.

Events D and E are exhaustive.

That is also true.

All the counters are either black or white, so those two events do not leave any counters out.

Events C and D are mutually exclusive.

That would be true.

In this case you can see that there are no counters that are both labelled C and are black, so that means that they are mutually exclusive.

The event C and D are exhaustive.

That would be false.

The reason why it's false is there are some counters that are neither labelled C or are black.

For example, there is a white counter labelled B, and there's another white counter labelled A.

Those counters are not accounted for by event C and D, so it's not exhaustive.

Events A, B and D are all mutually exclusive.

That statement would be false, and the reason why it's false is that there are some counters that are labelled A and are black, and there are some counters that are labelled B and are black.

So that means that there are counters that satisfy two or more of these events at the same time.

So all of these events are not mutually exclusive with each other.

And finally, as a statement A, B and D are exhaustive.

That would be false.

You can see that because there is a white counter labelled C and that counter does not satisfy any of those three events, so they are not exhaustive.

In question two, you've got a game where you spin the spinner twice and find the sum of the numbers.

And the table shows all the possible totals.

And you have to determine whether or not the events listed below are exhaustive and justify your answer.

For part A, the events were odd and even.

yes, these are exhaustive because all the possible outcomes are odd or even.

There are no numbers that are neither odd or even on there.

Part B, the event of getting a prime and a square number, they are not exhaustive.

There are some possible outcomes that are neither prime nor a square number.

For example, the number 6 is not prime and the number 6 is not a square number.

So 6 is not account for by either of those events, so those events are not exhaustive.

And part C, getting a factor of 60 or getting a number that is greater than 6, those events are exhaustive because all outcomes that are not factors of 60 are greater than 6.

So all outcomes satisfy at least one event.

You cannot find a number on that table that does not satisfy at least one of those events.

So yes, they are exhaustive.

D, the events where getting an odd number, getting a cube number and getting a square number.

Those events are not exhaustive.

There are some possible values that are not odd, not cube, are not square.

For example, the number two.

Then question three, you have to do the same thing again, but this time focusing on mutually exclusive.

So in part A, the events were odd and even, these are mutually exclusive.

A number cannot be both odd and even.

In part B, the events of getting a prime or a square number.

Yes, they are mutually exclusive.

A number cannot be both prime and a square number.

In part C, the events were factor 60 and greater than 6.

Well, no, they are not mutually exclusive.

It is possible to get a number that satisfies both of those events.

For example, the number 10 is a factor of 60 and it's also greater than 6.

That means they are not exclusive events.

And part D, the events were odd, cube number and square number.

Are they mutually exclusive? No.

For example, 9 is odd and square.

Yes, it's not a cube number, but the fact it satisfies more than one of those events means that all three of those events are not all mutually exclusive with each other.

You're doing great so far.

Now let's move on to the next part of this lesson where we're going to be summing probabilities to one.

Let's go back to the previous scenario where we had a regular six-sided dice, and the sample space shows all the possible outcomes when this is rolled.

Let's now consider the probabilities of some events.

The probability of getting an even number will be 3/6.

That's because three of the outcomes are even out of six possible outcomes.

The probability in an odd number would also be 3/6.

That's because three of the outcomes are odd out of six possible outcomes.

Let's now consider what would happen if we found these sum of these two probabilities.

We'd get 3/6 plus 3/6, which is one whole.

Now, the fact we get one whole is interesting, but why has it happened? What is it about these events that cause the probabilities to sum to one whole? Pause video while you think about this and press play when you're ready to continue.

These events sum to one whole because the events are mutely exclusive and exhaustive.

They're mutually exclusive because there are no outcomes which satisfy both events.

And they're exhaustive because there are no outcomes that are not accounted for by either of these events.

So we can see the probability of again even is 3/6.

There is 3/6 remaining while all of that is accounted for by the probability of getting an odd number.

Therefore because these are mutually exclusive and exhaustive, there are probabilities sum to one whole.

Let's now consider the probabilities of these two events.

The probability of getting a multiple of three would be 2/6.

There are two outcomes which are multiples of 3 outta six possible outcomes.

The probability of getting a multiple of 4 would be 1/6 because there is one outcome which is a multiple of 4 and there are six possible outcomes.

Let's now find the sum of these probabilities.

Get 2/6 plus 1/6 is 3/6.

These probabilities do not sum to one whole.

So why is that? Let's take a look.

These events are mutually exclusive.

There are no numbers that satisfy both of those events.

However, they are not exhaustive.

Therefore the probabilities do not sum to one whole.

So how about these two events? Getting a factor of 10 and getting a factor of 12.

The probability that we get a factor of 10 will be 3/6.

There are three numbers you can roll which are factors of 10.

There are probability and a factor of 12 will be 5/6.

There are five numbers you can roll which are factors of 12.

So let's take a look what happens when we find a sum of these probabilities.

We get 3/6 plus 5/6 is 8/6.

Again, they do not sum to one whole.

So why do these events have probabilities that do not sum to one whole? Perhaps pause the video while you think about this and press play when you're ready to continue.

Well, these events are exhaustive.

Every single outcome is accounted for but they are not mutually exclusive.

We can see that getting a one or getting a two would satisfy both of those events.

So those outcomes are accounted for in both of those probabilities.

That's why they do not sum to one whole.

So because the events are not mutually exclusive, the probabilities do not sum to one whole.

So let's check what we've learned.

We've got a bag which contains counters, label with letters from A to C.

Some counters are black and some counters are white and our counter is drawn at random from the back.

What is a probability that the counter is black? Pause video while you write it down and press play When you're ready for an answer.

The answer is 3/8.

There are three black counters and there are eight counters altogether.

What's the probability that the counter is white? Pause video while you write it down and press play When you're ready for an answer.

The answer is 5/8.

There are five counters which are white out of eight counters altogether.

So could you please find the sum of those two probabilities? Pause while you write it down and press play for an answer.

Well, we'll do 3/8 plus 5/8 and that'll give you one whole.

So why do these probabilities sum to one whole? Pause the video while you write down a sentence to explain why the probabilities for these events sum to one whole, particularly what is that about the events that mean they sum to one whole, then press play when you're ready to continue.

These events have probability that sum to one whole because they are both mutually exclusive and exhaustive.

So how about the probability of getting a counter with a C on it? What's that? Pause video while I=you write it down and press play when you're ready for an answer.

The probability of getting a counter with a C on it will be 1/8.

There's one counter with C and there are eight possible counters.

What's the sum of these two probabilities then? The probability of getting black and the probability of getting a C? Pause video while you write it down and press play when you're ready for an answer.

The answer is 4/8, which we get from doing 3/8 plus 1/8.

So these two probabilities do not sum to one whole.

Why is that? Pause video and write down a sentence or two which explains what is it about these events that means that probabilities don't sum to one whole and press play when you're ready to continue.

They do not sum to one hole because they are not exhaustive events.

The probability that an event happens and the probability that the event does not happen always sums to one whole.

For example, if we have a regular six-sided dice, which is rolled once we have all the outcomes displayed in this sample space.

The probability of rolling a multiple of 3 will be 2/6.

Draw two numbers which are multiples of 3 on a dice outta six possible numbers.

Let's now look at the probability that we do not roll a multiple of 3.

That would be 4/6 because there are four numbers on a dice which are not multiples of 3 out of six possible numbers.

And if we find that sum of these probabilities, we get 2/6 plus 4/6 which is one whole.

These two events are mutual exclusive and they are exhaustive.

They are mutually exclusive by definition.

You can't have a number which is both a multiple of 3 and not a multiple of 3 at the same time.

That's impossible.

So they're mutually exclusive.

And they're exhaustive because all outcomes which are not satisfied by one event are satisfied by the other events.

Therefore, the probabilities sum to one.

So for any event, the probability that it does happen and the probability that it doesn't happen sum to one whole.

That's because those two possible events are mutually exclusive and exhaustive.

Here's another example.

Alex plays a game and there are two outcomes at a game.

Either Alex wins or he does not win.

Now, those two outcomes are mutually exclusive and they are exhaustive.

So the probabilities sum to one.

And that means as the probability of one outcome increases the probability of the other decrease, like so.

Now, the fact that the probabilities of an event happening and not happening sum to one whole, it means that if we know the probability that an event happens, we can calculate the probability that the event does not happen by subtracting from one whole.

For example, if the probability that Alex wins the game is 0.

72, the probability that he does not win is 1 subtract 0.

72, which is 0.

28.

Now, this calculation could be performed with fractions, decimals or percentages.

For example, if the probability that Alex wins was 72%, we could calculate the probability he does not win by doing 100% subtract 72%, which gives 28%.

Or if the probability Alex wins is 18/25, we can calculate the probability he does not win by doing 1 subtract 18/25, which gives 7/25.

So let's check what we've learned.

Here, we have an event A, and the probability that A happens is 0.

4.

What's the probability A does not happen? Pause while you write it down and press play for an answer.

The answer is 0.

6.

How about with fractions this time? The probability that B happens is 5/7.

What's the probability that B does not happen? Pause the video while you write it down and press play for an answer.

The answer is 2/7.

Let's now work with percentages.

The probability that event C does not happen is 34%.

What's the probability that it does happen? Pause video while you write it down and press play for an answer.

The answer is 66%.

When we have situations where we know the probabilities sum to one whole, we may be able to use known probabilities to calculate unknown probabilities.

For example, here we have a spinner of letters, A to M.

And the table shows the probabilities of that the spinner lands on each letters, but there is a missing probability in that table.

Let's consider how we could use the known probabilities to calculate the probability that the spinner lands on D.

Perhaps pause the video and think about what approach you might take to this question and press play when you're ready to look at some solutions together.

Let's take a look at this together.

One method could be to calculate the probability that the spinner does not land on D.

And we can do that by finding the sum of 0.

17, 0.

23 and 0.

15.

They are the probabilities of all the outcomes that are not D.

If we do that we get 0.

55, and then to get the probability that it does land on D, we can do 1 subtract 0.

55 to get 0.

45.

A second method could be using algebra.

We could let x be the probability but it lands on D.

And we know that these four probabilities all sum to one whole, so we can write an equation that says exactly that.

We could then simplify that equation to get 0.

55 plus x equals 1, and then rearrange it to get x equals 0.

45.

We've done the same calculations as with the previous method and we can see the same numbers occurring.

0.

55 for example, was in both of those methods and the final answer was the same in each as well.

But the way we presented those calculations is slightly different.

Here's a slightly more complex problem where we can use known probabilities to calculate multiple unknown probabilities.

This time the table shows the probability that the spinner lands on B.

We don't know the probability it lands on A, C or D, but we can see a relationship between those probabilities.

The fact that both A and C have probabilities of x means those probabilities are equal to each other.

And the fact that probability it lands on D is 3x means that outcome is three times as likely as C or A, for that matter.

So let's find the value of x and then find the probability that it lands on each letter.

We could make an equation by saying the sum of all four of these probabilities is equal to one whole.

We could simplify the equation, then rearrange it by subtracting 0.

1 from each side and rearrange it some more by dividing both sides by five to get x equals 0.

18.

Now we haven't finished this problem yet.

We've got the value of x, but we need to find the probability of each outcome.

We can do that by using substitution.

The probability of A would just be one lot of 0.

18, which is 0.

18.

The probability of B was given to us.

The probability of C will be the same as A, which is 0.

18, and the probability D will be three times 0.

18, which is 0.

54.

So let's check what we learned.

A biassed dice is rolled and the table shows the probability of each outcome.

You can see that there's an empty space in fully probability of 4.

Could you please calculate the probability of rolling 4 with this dice.

Pause while you do it and press play for an answer.

The answer is 0.

09.

This time we have a similar scenario.

We can see that the probability of rolling 3 is 2x, the probability of rolling 4 is x, and the probability of rolling 5 is 3x.

Could you please calculate the value of x? Pause video while I do to it and press play for an answer.

The answer is x equals 0.

1.

And the work on the screen shows how you can get that.

So now we know that x equals to 0.

1.

Could you please calculate the probability of rolling 5 with this dice? Pause the video while you do it and press play for an answer.

The answer is 0.

3.

You can get that by substituting 0.

1 into x in the expression 3x.

And then this dice is rolled 1000 times.

Approximately how many times would you expect to roll 5? Pause video while you do that and press play when you're ready for an answer.

Well, if we take the probability of 0.

3 and multiply it by 1000, we get 300, which means after repeating the trial 1000 times, you expect to have got the outcome 5 approximately 300 times.

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

This task contains three questions.

and here is question one.

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

And here are questions two and three.

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

Okay, let's go through some answers.

In question one, you have to work out some missing probabilities.

A would be equal to 62%.

B would be equal to 0.

09.

C would be equal 9/13.

D would be equal to 0.

65 and E would be equal to 3/20.

Then question two, you have a table that shows the probability of each outcome when spinning a biassed spinner.

The probability that it lands an A would be 0.

1.

And if the spinner is used 1000 times, you'd expect that it would land on a approximately 100 times which you get from multiplying 0.

1 by 1000.

And question three, same scenario, but this time you have three unknown probabilities, but you do have a relationship between them presented as a form x, 3x and 4x.

To get the probability that the spinner lands in each letter you can create an equation and solve it to get the value of x is not 0.

125, which means the probability it lands in A is 0.

125, the probability lands and B is 0.

375 and the probability it lands on C is 0.

5.

And then if the spinner is used 5,000 times, approximately, you'd expect it to land on A 625 times, which you get from finding the product of the probability and the number of trials.

Fantastic work today.

Now let's summarise what we've learned.

Probabilities can be given as fractions, decimals or percentages.

Two or more events are mutually exclusive if they share no common outcome.

a set of events are exhaustive if at least one of them has to occur whenever an experiment is carried out.

And when events are mutually exclusive and exhaustive, the probabilities sum to one whole.

Great job today.

Have a nice day.