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Hi, I'm Mrs Dennett and in this lesson, we're going to be finding probabilities from frequency trees.

Here, we have some information about students in year nine and year 10.

Let's read through the information carefully together.

There are 256 students in years nine or 10.

Half of all the students are in year nine.

198 students participate in at least one extracurricular activity.

30 students in year 10, do not participate in any extracurricular activities.

A hundred students in year nine, do some form of extracurricular activity.

We're asked to draw a frequency tree and then find some probabilities.

We need to label our frequency tree, with years nine and 10 for the students.

And then with the other branches, participates for those who take part in an extracurricular activity, and doesn't participate for those students who do not do an extracurricular activity.

We can now fill in the numbers.

There are 256 students in total, and then we're told half of these are in year nine and half are in year 10.

Half of 256 is 128.

So we can fill this in.

We know that 30 students in year 10, don't do any extracurricular activities, whilst a hundred students in year nine, do some form of extracurricular activity.

We're told that 198 students do some extracurricular activity across both the age groups.

So as 100 are in year nine, the other 98 must be in year 10.

Finally, we need to work out how many nines don't participate.

This will be 128 takeaway a hundred, which leaves us with 28 students.

Now we have completed the frequency tree, we can find some probabilities.

Firstly, the probability of randomly selecting year nine students.

There are 128 students in year nine, out of a total of 256.

So 128 over 256 or a half.

Then we find the probability of randomly selecting a student who doesn't do any extracurricular activity.

There are 28 in year nine and 30 in year 10.

So altogether that's 58 out of 256.

Here's a question for you to try, pause the video to complete the task and restart when you're finished.

Here are the answers.

We don't have to draw the frequency tree in this question, we're actually given it.

So we just have to find the probabilities.

We want the probability that a person is randomly selected and is a man.

So there are 101 men out of a total of 204.

This gives us a probability of 101 out of 204.

Then we want the probability that we select a man who is under 25.

If we look at the top of the frequency tree, we can see that there are 38 men under the age of 25.

So this gives us 38 out of 204.

Finally, we want the probability if we select a person who is over 25.

We look on the all the 25 branches.

So for men, there are 63 over 25s.

And for women, there are 77 over 25s.

Add these together and we get 140 out of 204.

We're now going to look at a slightly different type of probability equation.

To do this we're going to use the same frequency tree that we used in our first example.

So we have our students in year nine and 10, and the frequency tree tells us whether or not they participate in extracurricular activities.

We want to find the following.

We're given the information that a year nine is selected, we can see that there are 128 year nines.

So we only want to concentrate on this part of the frequency tree.

We want the probability that if a year nine is selected, that they then do an extracurricular activity.

So we know there are 128 year nines, and we can see there are 100 students who participate in an extracurricular activity.

So we write 100 over 128.

Notice this time the denominator is not 256 because we have already been given the information that it's a year nine that is being selected.

Likewise, for part B, we're given the information that the student is in year 10.

And we want the probability, that we get a student who doesn't participate in an extracurricular activity.

So we're already told that the student is already in year 10.

So there are 128 students in year 10.

And we want to look at the probability that they don't participate in an extracurricular activity.

So this is 30 students.

So our probability will be 30 out of 128.

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.

So this is the answer to part A.

This is the frequency tree that we needed to draw, using the information given in the question.

Here are the answers to part B.

So if he wants to calculate the probability, that delivery is on time, we have to look for the on time section of the frequency tree.

And we can see that 44 deliveries were on time out of a total of eighty.

So the probability that they're on time is 45 out of eighty.

For part two, we want the probability that the delivery is accepted.

So here we need to look at two parts of the frequency tree, whether they were on time and accepted or late and accepted.

So this is 39 plus 32, which gives us 71 out of eighty deliveries.

Part three, we'd have to read this very carefully.

We want the probability that the delivery was accepted, given that it was late.

So we all need to look at the late part of the frequency tree.

We can see that 36 deliveries were late, and out of these 32 were accepted.

So the probability is 32 out of 36.

That's all for this lesson, thank you for watching.