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Hi, I'm Mrs. Dennett and in today's lesson, we're going to be finding probabilities from Venn diagrams. In this example, 50 children were asked whether they had brothers or sisters.

The results are shown in the Venn diagram.

We need to work out some probabilities.

Firstly, we're going to work out the probability that a child has a brother.

A child that has a sister, still has a brother, so, we need to include the overlap.

We need all the children in the circle labelled brothers.

So, we add up 14 and 18 to give 32.

There are 50 children in total.

So, we write 32 out of 50, which simplifies to 16 out of 25.

Now we're going to work out the probability a child doesn't have a sister.

We can ignore the circle labelled sisters and add up all of the other numbers.

14, add three is 17, out of a total of 50.

So, the probability that the child doesn't have a sister is 17 out of 50.

You may notice that I've used the dash notation here.

Remember, that means the complement, so, sister dash means the complement of sister or not having a sister.

Now, we want to work out the probability that a child has a brother or a sister.

This includes the overlap.

It includes children who have a brother and a sister.

This is represented by this shaded section, which is also called the union of the sets.

We need to add together all these numbers.

So, the probability that a child has a brother or a sister is 47 out of 50.

Notice the symbol for union being used here, a shorthand for all, brother or sister.

Finally, we're going to find the probability that a child has a brother and a sister.

This time, we look at the overlap of the Venn diagram.

This represents children that have both a brother and a sister.

And is also called the intersection of the two sets.

There are 18 children that have both.

So, the probability is 18 out of 50, or nine over 25, when simplified.

Notice the intercept notation being used here, a shorthand for and, brother and sister.

Here's a question for you to try.

Pause the video to complete the task and restart when you are finished.

Here are the answers.

Firstly, we need to draw a Venn diagram.

We start by filling in the intersection, that is the overlap, where children who play cricket and badminton go.

There are seven children who do both, so seven goes here.

We're told that nine children play cricket.

Seven have already said they do both, which leaves us with two children, who only play cricket.

We can't fill in the remaining parts of the badminton circle yet, but we are told that three children play neither.

So, we put these children inside the rectangle but outside of the two circles.

As we're told that there are 15 children altogether, we can add up three and two and seven to get 12 and find that there are three remaining children in the universal set.

So, these children must go in the remaining part of the badminton circle.

For part B, we look at the badminton circle only.

There are seven plus three children here.

So, the probability that a child plays badminton is 10 out of 15, which can be simplified to two thirds.

Here is a question for you to try.

Pause the video let's complete the task and restart when you are finished.

Here are the answers.

For part A, we want the probability of A union B.

This is all of circle a and all of circle B, including the overlap.

There are six numbers in this section, so, the probability is six out of eight.

As there are a total of eight numbers in the universal set.

Then we need to find the complement there.

Ignore anything that is in circle A.

Refer members not in A.

So, the complement of A is four eight or a half.

Here is a question for you to try Pause the videos complete the task and restart when you are finished.

Here are the answers.

The first statement is false, because in A intersect B, there is only one member.

That's the overlap.

So, the probability here should be one ninth.

This means that the second statement is true.

There is only one number in the intersection.

We're now going to take a look at a Venn diagram with more than two circles.

In this question, we have three sets, A, B and C.

We want to firstly work out that A, union with B ,union with C.

Let's look at this region on the Venn diagram.

We want all the members of A, all the members of B, and all the members of C, this includes the overlaps.

I add up all of the numbers in A, union B union C, and get 30.

In total, there are 40 members as there is a 10 outside of the circle, well, or three circles I should say.

So, my probability will be 30 over 40, or three quarters.

Next, I want to work out A intersect B, intersect C.

This is where all three circles overlap.

I want the members in A and B and C.

The only members are right in the middle.

There are three, so, the probability is three out of 40.

How can we work out the probability of the complement of A and B, and C.

We saw in the previous question, that there were three members of A intersect B, intersect C.

If there's 40 members in total, and there must be 37 members in the complement of A and B and C.

You should also remember that the sum of probabilities is one.

So, one take away three over 40 leaves me with 37 over 40.

Now, we're going to work out probability A intersect C.

This is where circle A overlaps with circle C.

There are four numbers in this region, so, the probability of A intersect C is four over 40 or one tenth.

Now let's look at B union C.

This is everything in B, or everything in C.

In total there are 21 members of B union C, five, add six, add three, add two, add four, add one, which gives us 21, so, the probability is 21 out of 40.

Finally, we want to work out the complement of A union B union C.

This means everything that isn't in that union.

From the diagram, we can see that there are 10 elements not in the union, giving us 10 out of 40 are a quarter.

Alternatively, in part A, we worked out A union B, union C was 30 out of 40, or three quarters.

We can work out the complements of A union B, union C, using the fact that probabilities sum to one.

So, one take away 30 equals 10 40th or a quarter.

Here's a question for you to try.

Pause the video to complete this task and restart when you are finished.

Here are the answers.

For part A, we look at the overlap for geography and history.

We can see that six students do geography and history and art, and eight do geography and history.

So, we get 14 out of a total of 27 students.

Similarly, for part B, students who do art and history, we have three and six.

So, the probability that students studies art and history is nine out of 27.

Finally, for Part C, we are looking for the overlap of all three sets.

This is right in the middle.

So, we can see that there are six students.

So, the probability is six out of 27.

That's all for this lesson.

Thank you for watching.