Lesson video

Hi, I'm Mrs. Wheelhouse and welcome to today's lesson on experiments to determine how likely an outcome is, and this lesson is from the unit, Probability: possible outcomes.

I love probability.

So, I'm really looking forward to today's lesson.

Let's get started.

By the end of today's lesson, you'll be able to determine the likelihood of an outcome by designing and carrying out an experiment.

So, the key word that we're going to be using today is the word experiment.

And what we mean by this in terms of probability is an experiment, is a repetition of a trial multiple times in order to observe how often each outcome occurs.

So in a way, an event could be just one outcome or a group of outcomes.

An experiment can be thought of as a trial or a collection of trials, depending on how many trials make up an experiment.

Our lesson comprises of two parts today and we're gonna start with part one, which is on planning and conducting experiments.

When flipping a coin, there are two possible outcomes.

The coin lands showing heads or the coin can land showing tails.

Now we are of course, assuming that the coin lands flat, so it is possible to flip a fair coin and have it land right on its edge, but it's really unlikely.

So, for our experiment, we are just saying there are two outcomes, heads or tails.

So, we can see that there.

Our trial is to flip the coin once and the set of outcomes includes heads and tails.

Now, each outcome is equally likely to happen, because these are the only two possible outcomes.

The likelihood of each outcome is an even chance and this is because the coin isn't weighted a particular way.

Now, Jacob and Sophia are playing with a Frisbee, and Jacob says, "If I flip this Frisbee in the air, "there are two possible ways that it could land." So, we could have it land so that the open part of the Frisbee is facing down or the open part could be facing up.

Do you think the two outcomes are equally likely? Sophia says the shape of the Frisbee, means that it's two sides are different, so it could be more likely to land one way than the other.

How could they investigate whether one outcome is more likely than the other? What do you think? Well, Jacob says, "To make it easy to talk about the outcomes, "we could call one outcome heads "and the other outcome tells "and you can see what we're defining as heads and tails "on the screen right now." Sophia says, "We could flip the Frisbee in the air several times "and see which outcome occurs the most.

"That will give us an indication "as to whether one outcome is more likely than the other." So, they flip the Frisbee in the air 10 times and record the results.

And you can see them here in our bar chart.

The results were four heads and six tails.

So Jacob says, "We can conclude the Frisbee "is more likely to land on tails." Is that right? Sophia says, "Well, possibly.

"But this experiment hasn't really convinced me." Has it really convinced you? Do you think flipping it in the air 10 times is enough to say that tails is more likely than heads? How could they be more confident about their conclusion? Pause, have a quick discussion and then start the video again.

Welcome back.

What did you come up with? Well, in this case, Jacob and Sophia are going to do more trials.

They're gonna do more and more.

They're gonna keep experimenting.

Well in Jacob and Sophia's case, they're now gonna flip the Frisbee 20 times and record their results.

Well, this time there were 13 heads and seven tails.

Wow, this experiment suggests a different conclusion that the Frisbee is more likely to land on heads than tails.

Sophia says, "These results seem a little more trustworthy, "but I'm still not fully convinced." Are you? What could they do to be even more confident about their conclusion? Wow, they flipped the Frisbee in the air 500 times and record the results.

And you can see the results here.

The results were 281 heads and 219 tails.

Does this mean we can conclude the Frisbee is more likely to land on heads than tails? And Sophia says, "Well, we don't know for absolute certain, "but these results could be convincing enough." Now if you want, you can click on the link for the Desmos simulation and you can run the Frisbee flip simulation.

Here is our Frisbee flip simulation.

In order to run it, all you need to do is just click the Run button.

Let's see what happens.

And you can see I flipped the Frisbee 500 times again, can see that there? And you can see once again, heads came up more often than tails.

So, it's seeming likely that that's a suitable conclusion.

An experiment is a repetition of a trial multiple times in order to observe how often each outcome occurs.

If one outcome has a greater frequency than the others, it could suggest that that outcome has a greater likelihood of occurring than the others do.

We might not be able to be absolutely certain that our conclusions from an experiment are true.

And we saw that, didn't we? With Jacob and Sophia when they only flipped the Frisbee 10 times and even 20, they weren't absolutely certain.

In fact, they weren't even pretty convinced, it wasn't until they did it 500 times that they're starting to be more convinced, but they're still not absolutely certain.

The more trials and experiment contains, the more confident we can be about any conclusions, which is why we were more confident when we were doing 500 trials and a lot less confident was it was only 10 or 20.

Jacob and Sophia represent the conclusion from their Frisbee experiment on a scale of likelihoods.

So, you can see here where they've put tails and where they've put heads of coming up.

"That doesn't look right though," says Jacob.

Why must at least one of those crosses be in the wrong position? That's right, fewer than half of the trials resulted in tails.

Remember, we conducted 500 trials.

So, how can tails be likely? Tales therefore has to go, before the halfway point of the scale, and this means that tails has a less than even chance.

The more likely the Frisbee is to land on heads, the less likely it is to land on tails.

Let's do a quick check.

A toy building brick that is thrown into the air, could land in one of four possible ways.

How could you investigate, which outcomes are more likely than others? Pause the video and write down all the different ways you think you might be able to investigate which outcomes are more likely than the others.

Welcome back.

What did you put? Well, I suggested that you could conduct an experiment by throwing the toy brick into the air multiple times.

So, let's conduct an experiment where we throw that toy brick in the air a hundred times.

The results are shown in our bar chart.

The toy brick is about to be thrown in the air again.

Based on these results, which outcome do you think, would be the most likely to happen? Pause and write down what you think.

Which one did you go for? Personally, I'd have gone for B.

It had the highest frequency, so it seems likely that that's gonna happen again.

True or false, none of the events have a likelihood that is more than even chance.

Is that true or false? Once you've decided, you then need to justify your answer by selecting either A or B.

Pause and do this now.

So, what did you go for? Well, you should have said that this is true.

So, none of the events have a likelihood that is more than an even chance.

And how are we justifying that? Well, it's because none of the events have a frequency that is greater than half of the number of trials.

Remember, the number of trials was a hundred.

So, to have a likelihood that's more than an even chance, we needed to have the frequency be more than 50.

The markings on the scale, represent the likelihoods of each event.

Which event do you think is indicated? So, which one is inside the circle? Would it be event A, B, C, or D? Pause the video and make your choice now.

Which one did you go for? I went for C.

If we think about it, the least likely of those events is A, so that's gonna be closest to impossible.

Then the next tallest bar is D, so that's the next cross.

C is the next tallest bar.

So, that must be our third cross.

And then closest towards the even chance, will be the one with the greatest frequency.

And that's B.

It's now time for your first task.

For question one, we're talking about dropping a cone and it can land in two ways.

It can either land base down, which you can see is called A or curve down, which we're calling B.

In part A I'd like you to plan an experiment to investigate which outcome is most likely to happen.

In part B, the bar chart shows the results from an experiment that I carried out and the cone is about to be dropped again, in which position is it most likely to land, based on these results.

And then C, please use the link to access the simulation of this experiment.

Press the Run button to begin the simulation and the Reset button if you wish to start again.

Try this multiple times with a large number of trials.

Do those results affect your answers to part B? Pause the video while you do this now.

Welcome back.

Let's look now at question two.

So, we've got our toy brick again and we are throwing it into the air and it can land in one of four possible ways.

In an experiment, we throw the toy brick 20 times and produce the following results.

For part A, the toy brick is about to be thrown again, draw a scale of likelihoods and use the results from the experiment to show the likelihood of each outcome on the scale.

Then in part B, use the link to access a simulation of this experiment.

Again, press the Run button to run the simulation and reset it if you need to.

Run the experiment multiple times, remember we want a large number of trials.

Do those results affect your answers to part A? Pause the video while you have a go at this.

Welcome back.

How did you get on? Let's go through some answers.

So for part A, you had to plan an experiment and you could have said, "I'm going to drop a cone 50 times." So you can do 50 trials.

You could say, "I'm gonna drop a cone onto the floor "from the same height 200 times "and record the number of times it lands in each position." That second answer is definitely more precise than the first answer.

I think it's a much better answer.

Now, the important thing is that your number of trials you suggested could easily vary, but the larger the number of trials, the more likely you're going to get results that you can perhaps believe in a little bit or maybe trust a little bit more.

But you do need to consider the practical limitations.

So, if you've said you're gonna drop this cone a million times, think about how long that's going to take.

It might not be practical.

In B, we showed you the results for an experiment and we're gonna drop it again, in which position is the cone most likely to land? And you should have said A, which was based down because that has a higher frequency.

And then in part C, we said have a go at the simulation.

Run it multiple times.

Do the results affect your answers to part B? What you may have said or you may have noticed is that when using a large number of trials, the cone lands curve down more often than base down.

So, the curve down appears to be the more likely outcome.

Now of course, you may have seen different things, depending on what results you got from your simulation.

And in question two I said, please draw a scale of likelihoods and mark where you think A, B, C, and D, should be on that scale.

And it should look like this.

Remember, we threw the brick in the air 20 times and the highest frequency was B, which looks to be about eight.

In other words, it's under half of the number of trials we did.

That means that B cannot be greater than an even chance.

And so you can see where I've put the events there on my scale of likelihoods.

Then I said, please have a go at running the simulation multiple times with the large number of trials.

You may have noticed that when using a large number of trials, for example, a thousand.

Outcome C happens the most often.

So, it appears to have the greatest likelihood, outcome A happened the least often, so it appears to have the least likelihood.

But again, your answer may vary, depending on what you saw in your simulation.

It's now time for part two of the lesson and in part two, we're gonna be considering the number of trials required.

I wonder what I mean by that.

Let's find out.

Laura and Andeep are playing game that involves rolling a six-sided dice.

Laura says, I don't think this dice is fair.

It keeps landing on some numbers more than others.

Perhaps, it's not weighted equally throughout the dice.

Andeep it says, "Well, we could investigate whether the dice is fair "by running an experiment." True, how many times though, do you think they need to roll the dice for their experiment? Do you think rolling it maybe six times is enough? Pause the video and have a discussion about what you think? What did you say? Laura and Andeep decide they're going to carry an experiment by rolling the dice 120 times.

So, they think 120 trials is enough.

And you can see the results here.

Laura says, "I told you that the dice wasn't fair.

"The frequency for five is far higher than the others." And it says, but even when events have an equal chance of happening, one event can happen more often than another.

Well, let's do the experiment again.

So, they again do another experiment rolling the dice 120 times and now this is the results they get.

Oh, the results do look quite different.

Now, Laura's not sure if the dice is fair.

Andeep goes, "Well, maybe 120 trials isn't enough, "because we ran our experiment twice "and we got differing results." Maybe we need to do a lot more trials.

So, now this time they roll the dice 1,200 times.

Luckily they have a lot of friends and their friends can help them out.

And this is the results they get.

Laura notices that the more trials they ran in the experiment, the more it looks like the numbers on the dice are equally likely.

And it says, "I wonder why using more trials, "means the dice looks to be more fair." Now, even when events are equally likely, they don't necessarily occur, an equal number of times during an experiment.

One event may occur more often than the other and that's just by chance.

For example, when flipping a fair coin, the two outcomes heads and tails are equally likely.

But you can still get more heads than tails.

For example, have a look at our graph.

You can see we flipped the coin a hundred times and we saw 60 heads and 40 tails.

If you like, you can click on the link below and we've produced you a fair coin simulation.

So, saving you having to flip the coin, all that number of times, our simulation will do it for you and you can move the slide to see results from different experiments.

Now, even when one event is more likely to happen than another, the event that is less likely to happen may occur more often during an experiment.

For example, when flipping that Frisbee, we said it looked like it'd be more likely to get heads than tails because of the shape of the Frisbee, but you can still get more tails than heads.

So, we tried out this experiment and these are the results we got.

And if you've been playing with that simulation on the Frisbee, you might have seen something like this too.

An unlikely event can happen more often than a likely event simply by chance.

This tends to happen more often in experiments that have a very small number of trials.

The more trials an experiment contains, the more likely we are to observe an accurate picture of the likelihoods for each event.

So, the more trials an experiment contains, the more confident we can be with any conclusions drawn from the experiment about whether one outcome is more likely than another.

And as you saw, we can run multiple experiments to see if we get similar results each time.

But experiments with a small number of trials can produce vastly different results.

And that's how we can suggest to ourselves that perhaps we are not seeing representative results.

So for example, let's roll a dice 60 times.

I did this experiment twice and you can see the two different sets of results I got.

They are quite different.

If you wanna see some more simulations of experiments with a fair dice, you can click on the link we've put down below.

Now if you move the slider, you can see all the different experiments we did and you can see how the results vary greatly from one experiment to the next.

Experiments with a large number of trials are more likely to produce more consistent results.

For example, when we roll the dice 6,000 times, I did that twice.

I did use my simulation and this is what you can see.

They're a lot closer, so far more similar now, and again, if you want to have a go at seeing what happens if you roll the dice 6,000 times, you can click on the link here and be taken to a Desmos simulation.

You'll notice as you move the slider for K and see all the different experiments we did that the results are far less varied.

In other words, the bars are more consistently the same.

Time for a quick check now, if event A is more likely than event B, then A will always happen more often than B, during an experiment.

Is that true or false? Don't forget to justify your answer by selecting either A or B.

Pause the video while you do this.

Welcome back.

Now you should have said false.

And what did you pick to justify? It should have been A, an event that has a lower likelihood, may still happen more often and that's by chance.

What about this question? If event A has a greater frequency than event B during an experiment, you can be certain that A is more likely than B.

Choose whether that's true or false and don't forget to justify your answer.

Pause the video while you do this.

Welcome back.

You should have selected false.

You definitely can't be certain, because an event that has a lower likelihood, may still happen more often by chance.

What about this check? How can an experiment be improved to create a more reliable set of results? Could you conduct an experiment with a greater number of trials? Conduct an experiment with a smaller number of trials or an experiment can never be improved? Which one do you think? Pause and select now.

Welcome back.

You should have picked A.

The greater number of trials is likely to create a more reliable set of results.

It's now time for our second and in fact, final task.

We're gonna start with question one.

Where a class is investigating which outcome is most likely when we flip a Frisbee.

In A, an experiment conducted with 10 trials and you see the results.

Does one outcome appear more likely to happen than the other? And if so, which one? For B, use the link to access the simulation of this experiment.

Press the Run button to begin it and let it run until the count shows 500, because that's 500 trials.

Do those results differ from the experiment in part A? And then C, which experiments results seem more reliable to you? Don't forget to justify your answer.

Pause the video while you complete this question.

Welcome back.

Let's go through our answers.

So, based on this experiment with 10 trials, does one outcome appear more likely to happen than the other? The answer is no.

They appear equally likely.

We did 10 trials and each outcome came up five times.

Now, if you used the simulation and had a look, did you write down what you noticed about the results from the experiment in comparison to what happened in part A? So, you might well have noticed that when using a large number of trials, heads happen slightly more often than tails and this could suggest that heads is slightly more likely to come up than tails is.

And which experience results seem most reliable to you? While I would say part B, because it uses a greater number of trials.

In other words, the chance is having less of an impact.

Remember, it is possible that the event that's less likely came up more often if we have a small number of trials.

So, the greater the number of trials, the more reliable our results are likely to be.

Time now for a quick summary.

An experiment can be designed to investigate whether one event is more likely than another.

An experiment is a repetition of a trial multiple times in order to observe how often each outcome occurs.

The results from an experiment can be used to determine the likelihood of each event and a large number of trials may be needed in experiment to produce more reliable results.

Well done.

I hope you had a great time in today's lesson and you enjoyed using those simulations.

Of course, you didn't have to use them.

You could have carried out the experiment yourself, but personally I wouldn't wanna have to roll a dice 6,000 times.

I'm really grateful that technology can help me out here.

I look forward to seeing you for more of our lessons on probability.