Lesson video

Hello, I'm Mr. Langton.

And today we're going to look at theoretical probabilities and relative frequency.

All you're going to need is something to write with and something to write on.

Try and find a quiet space where you won't be disturbed.

When you're ready, we'll begin.

We'll start with the try this activity.

The spinner is spun 60 times.

How many times would you expect to score a one, a two, and a three? How would this change if you spun the spinner a 100 times? what about 10 times? Pause the video and have a go.

When you're ready, unpause it and we can go through it together.

You can pause in three, two, one.

Okay, let's go through it together.

If I spin the spinner 60 times, how many times would I expect it to land on one? Well, looking at what we've got, there are six possible options and three of them are one.

The probability of getting a one, I'm going to write it like this, the probability that I score a one, will be 3/6, which is a 1/2.

So if I spin it 60 times, half of the time, I'm expecting it to come up as a one.

So half of the time will be 30 times.

Now what about two? Let's have a different colour.

Two of the possible outcomes are two, the probability of scoring a two, is going to be 2/6, and you think well to a 1/3.

And if I do it 60 times, a 1/3 of 60 is 20.

So if I spin it 60 times, and I've got 30 20 is 50.

We can be pretty confident already that this answer will be 10, but it wouldn't have to check would it? What's the probability that I score three? There's one way to get a 3/6.

On 1/6 of 60 is 10.

Now for the next part, how would this change if you spun the spinner a 100 times? What we're going to get more of each number aren't we? But ideally they should stay in proportion.

If I did it a 100 times, then to get a one half of the time, it should come up with one.

So that'll be 50 times for a one and a 1/3 of the time.

Now a 1/3 of a 100, we're going to get a decimal there.

We're going to get 33.

33333.

I'm just going to round it to 33.

And a 1/6 of the time is 16.

6666666.

So that's going to around to 17.

So that's roughly what we would expect if you did it a 100 times.

If we only did it 10 times, we said half the time it should come up with a one.

Now a 1/6 of 10, sorry, a 1/3 of 10 to the profit for two a third of 10.

It's going to be 3.

33333.

So we're expecting about three times.

But it's a decimal and it's been around but so few attempts, 3.

333 is, well, isn't going to copy exactly three times.

It's starting to get a little bit messy now.

Cause here the thing, the property comes up with landing on a three.

That's just one chance out of six.

If we do it 10 times, we should be looking at 1.

6 recurring.

So between one time and two times, and ideally if you want it to, it should add up to 10.

So ideally we're looking at two, but I've started to see now this is looking a little bit weird.

I'm not entirely confident in my answers there.

It's as close as I can get, but it just doesn't feel right.

Does it? We can try and use probability to make predictions on how many times something will happen.

What we start with is the actual probability of what we think it will be.

And that's, what's called the theoretical probability.

This is what I've been working with for the last few lessons.

So the theoretical probability of scoring a one, we said there were three ways that we can get a one.

So the probability of getting a one is three out of all the possible options, 3/6 or a 1/2 of 0.

5 The theoretical probability of scoring a two, probability of scoring two.

We said it's 2/6 or a 1/3, which is 0.

3 recurring.

And the theoretical probability of scoring a one, I'm trying to write like that.

No problem.

So , I messed that one up.

Didn't? The theoretical probability of scoring a three, there is one way to do it.

And 1/3 and 1/6 is 0.

16 recurring.

And we call those the theoretical probability is that's what should happen.

Now, once we start doing an experiment and we test it, we call that the relative frequency.

So in this experiment, I spun the spinner a hundred times and I actually came up 43 times on a one.

So the relative frequency is 43 amps at a 100.

I did 40.

I did a hundred trials and 43 times it came up with a one.

So the relative frequency is 0.

43 I've got two 37 times.

That's 37 out of a hundred, 0.

37.

And there are 20 different times it came with a three.

20 out of a 100 is 0.

2.

So we can compare the results from our experiment with a theoretical probability of what should have happened.

So the theoretical probability of scoring a one was 0.

5, exactly a half.

Now in reality when we did the experiments, we've got 0.

43.

So it's close, but it's not spot.

Theoretical probability of scoring a two was exactly a 1/3 of 7.

3 recurring.

Now but there's a 0.

37 here, it's pretty close.

So once again, we can say we're pretty much on track.

It's matching up roughly with what we've got.

Not exactly, but roughly.

Now the probability of scoring a three, 0.

16 recurring.

That came up a little bit more often than, than we expected, 20/100 or 0.

2.

And the idea, the thinking behind these probabilities is, when you do the experiment, it won't always exactly match up to what you want.

But ideally the more trials you do and the more experiments you do over time, it will start to get closer and closer.

Think about that last slide, when we said we did it 10 times, we're expecting it to come up with a three, 1.

6 times, which is a little bit, it can't happen 1.

6 times.

So it's either going to be lower than we expected or higher than we expected.

But as we do more and more trials, it's more likely that we'll get an experimental probability that is much closer to the theoretical probability.

Okay now it's your turn.

Pause the video and have a look at the worksheet.

When you're ready, unpause it and we can go through it together.

Good luck.

How did he get on? Let's look at the answers.

Cala decides to throw a coin 50 times.

She says "I will get 25 heads and 25 tails".

Explain why she could be wrong.

Well the reason that she could be wrong is that she's doing an experiment and it might not exactly match what she's got.

She might find that she gets 30 heads and 20 tails.

Maybe it's 26 heads and 24 tails.

It's not always certain that he's going to come out exactly the way that you expect it to.

There's always an element of chance a probability that could come into it.

The more experiments she does, the more likely it is to result in an even 50-50 split.

But when you're looking at things that precise, there's a chance it's going to be out by a tiny bit either way.

And the more experiments you do the better.

Next part.

I roll a dice 30 times and I get these results.

But complete the relative frequency table in order to see if we think the dice is biassed or not.

So there, he came up with a one twice and we did 30 trials.

So the relative frequencies 2/30, it came up with two twos out of 30.

Number threes there are seven threes up to 30.

Number of fours, four comes up three times out of 30.

Five only comes up once.

And six comes up 15 times.

Now the first thing that I want to do is I'm going to check.

It says that I've done this experiment 30 times.

Let's check how I got our counter up to 32.

Two and two is four out seven is 11, 12, 13, 14, 15, 13.

Yes! Right okay.

So I completed the table.

Do you think the dice is biassed? Well, if we look at it, the chance of getting a one or a two are pretty similar, and so you have the same for the four and the five.

There's not that much difference between them.

The three things, quite a bit more likely seven times out with 30 instead of two or three, but that's six there.

That is what is really frustrating me.

It stands out quite a lot.

Half of the time it's coming out with a six.

If I've got six options and half of the time it's coming up with a six, I'm starting to think that, yes, he's biassed.

I finally rolled it a few times.

Say I rolled the dice twice and I got two sixes then, that's not very likely, but it could happen that doesn't say that it's biassed.

If I rolled it five times and got a couple of sixes, yeah, that's quite reasonable.

If after 30 trials and half of the time it's coming up as a six, half of the rest of the time, it's split between five other options that doesn't feel right to me.

And I feel absolutely, no, I think it probably is biassed.

Okay and now for the next part.

I tossed a coin 200 times and I get 120 heads.

What is the relative frequency of getting a head? So I'm doing it as an experiment and the experiment it came with a head 120 times out of 200 trials.

If we like, we can cancel that down.

It would cancel out to 60/100.

Which is really useful for me because I know that that's going to be 60% or 0.

6 and keep cancelling that fraction down to 3/5.

What is the relative frequency of getting a tail? So we can do that two ways.

We can say, well, okay, if it came to a head 120 times, then it's going to come just tails 80/200.

And we can cancel that down.

We could look at the probabilities we've already got.

If the probability, the relative frequency of getting ahead is 3/5, relative frequency of getting a tail must be 2/5, make a whole one that the only possible two options are complimentary.

The relative frequency is 60% for a head.

So it must be 40% for a tail.

And 0.

6.

give, us 0.

4.

Finally, I roll a dice 60 times.

The number four comes up eight times.

Is the relative frequency higher or lower than the theoretical probability? So the relative frequency, if I've done it eight times, then that is 8/60.

Now the theoretical probability of getting a particular number, we're getting a four would be one six, 1/6.

If I do it 60 times, I would expect it to come up with a four 10 of those times.

So in my experiment, it came up slightly less likely than normal.

So it is lower.

The relative frequency is lower than the theoretical probability.

Okay now it's time for the last activity.

This one is quite a tricky one.

The spinner with 16 segments was shaded in red, yellow, and blue.

After 100 spins, the relative frequency on the right was recorded.

The colours were then removed from the spinner.

So you couldn't see, which was which.

How many segments do you think were shaded in each colour? And what would the theoretical probability of being through each one? Pause the video and have a go.

When you're ready, unpause it and we can go through it together.

So pause in three, two, one.

Okay.

Let's start and have a look at what it could have been.

Now for the relative frequency of getting a red is 0.

55.

So that's telling me that 55% of the time it came up with red.

So I would expect 55% of the options to be red.

55% of 16 is 8.

8, which is pretty much nine.

So I'm going to guess that nine of them are red.

And that gives me a probability, the probability of getting red is 9/16, which happens to be 0.

5625.

Right, next to yellow.

It came up as yellow 18% of the time.

And 18% to 16 is 2.

88.

So I'm going to round that.

I'm going to round that to three, I think, yeah, I'm going to round it to three.

The probability of getting yellow is going to be 3/16.

Now blue.

It came up blue 27% of the time.

27% of 16, cause there are 16 options is 4.

32.

I'm going to run that down.

I'll run that to four.

So the probability that I get a blue is 4/16.

Now looking at what we've got, number of outcomes nine, add three add four does make 16.

So that is one possible way that this spinner could have been shaded in with nine reds, three yellows and four blues, that colours in all 16 segments.

And that would give us probabilities.

It's really close to the relative frequency.

So I can be fairly confident that I probably got a decent answer there.

Right, we're going to leave it there for today.

Well done.

I'll see you next time.

Goodbye.