Lesson video

Hi there.

My name is Dr.

Olson and I'm excited to 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 Experimental Probability, and by the end of today's lesson, we will be able to decide whether a game or object is fair and calculate the experimental probability.

This lesson will introduce two new keywords.

One of the new keywords you guessed it, is experimental probability.

An experimental probability can be determined by the number of times an event occurred during an experiment, and we'll see plenty of examples of that during today's lesson.

The other key word is the word fair.

Now, words like fair and fairness are commonly used day to day to mean a very similar thing to what we're going to talk about in today's lesson.

In the context of probability, a fair experiment is where each outcome has the same chance of happening.

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

You may want to pause the video if you need to remind yourself what any of these words mean, and then press play when you're ready to continue.

The lesson is broken into two learning cycles.

In the first part lesson, we're going to be using experiments to test furnace to simply get a sense of whether or not two or more outcomes are equally likely to happen or whether one the outcomes is more likely than the others.

And then the second part of lesson, we're going to be on a bit further by looking at how to calculate experimental probabilities based on the results of experiments and then use those probabilities to do other things.

Let's start with using experiments to test fairness.

Let's start with the scenario of flipping a coin.

When flipping a fair coin, there are two possible outcomes.

The coin could land showing heads or it could land showing tails.

Each outcome here is equally likely to happen.

Therefore, we can find theoretical probabilities for each outcome.

The probability of both outcomes would sum to one, and if those outcomes are equally likely to happen, we can calculate the probability by doing one divided by two to get a probability of one half for each of those outcomes.

Let's take a second to consider, why are these two outcomes equally likely to occur? What is it about coins that means they are equally likely to land on heads as they are on tails? Perhaps pause the video while you think about this and press play when you're ready to continue together.

Well, the thing about coins is that the shape is symmetrical between heads and tails and the weight is pretty much uniformly distributed throughout the coin, and that's what makes it so they are equally likely to land on either other sides.

Let's now think of a different scenario which involves flipping a frisbee.

Quite often games and sports that involve frisbees will start by someone flipping a frisbee in the air to help decide who should go first in that game.

When flipping a frisbee in the air, it could land in one of two ways, and these are referred to as heads and tails.

You can see a diagram illustrating those on the screen.

Now, the shape and weight of a frisbee means that the outcomes here are not necessarily equally likely.

One of those may be slightly more likely to happen than the other but because the shape of the frisbee, we cannot necessarily calculate the exact probability that it could land in heads or on tails.

Therefore, a theoretical probability cannot be found.

So if we can't find a theoretical probability by dividing one by two for example, how could we determine which outcome is more likely to happen than the other? Pause the video while you think about this and press play when you're ready to continue.

Well, one thing we could do is conduct an experiment to try and find out which one is more likely to happen than the other.

Let's unpack experiments a little bit now.

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 could have a greater likelihood of occurring than the other outcomes.

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

Sometimes two things can be equally likely to happen, but one will happen more than the other simply by chance.

And sometimes the thing that is less likely to happen might happen more than the thing that is more likely to happen simply by chance as well.

So an experiment can give us a good sense of which one is more likely to happen than the other, but we cannot be absolutely certain from its results.

However, the more trials that an experiment contains, the more confident we can be about any conclusions we make based on its results.

Let's unpack the word fair now.

A trial in our experiment is fair if each outcome has an equal chance of happening.

Each outcome is said to be equally likely, and we can see some examples of fair spinners on the screen.

For example, the one with just A and B on, that spinner is equally likely to land an either A or B because the sectors are the same size and the same can be said for the spinner with A, B, and C on.

They are all equally likely as each other to happen.

If there is bias in a trial, the outcomes are not equally likely.

And here are some examples of bias spinners.

We can see that with a spinner with just A and B, that spinner is more likely to land on A than it is on B, because the area for A is bigger than the area for B.

And we can see with spinner that the those three outcomes are not all equally likely to happen.

It's probably more likely to land on B than either C or A.

Now, it's not always as easy as this to see whether or not it trial is fair or biassed.

Therefore, an experiment can help us to investigate this.

Let's take a look at an example.

Here we have Jun who conducts an experiment where a frisbee is flipped 10 times and here are his results.

The frisbee landed on tails more often than it landed on heads.

So what conclusion could Jun draw from this experiment? Perhaps pause the video while you think about this yourself and press play when you're ready to hear from Jun.

Let's hear from Jun.

He says, "On one hand I might infer that the frisbee is always more likely to land on tails." Because that's what's happened in this experiment.

It's landed on tails more than heads.

On the other hand, even if the outcomes were equally likely, this could still happen just by chance.

So how could Jun be more confident about his conclusion? Pause video while you think about that and press play when you're ready to continue.

Well, one thing you could do is run the experiment again.

Jun repeats the experiment where a frisbee is flipped 10 times and this time the frisbee landed on heads more often than it landed on tails.

And we can see that in the bar chart.

Jun says, "The results from this experiment could suggest that heads is more likely.

But again, this can happen just by chance.

This experiment doesn't seem very convincing." So how could Jun improve this experiment so that his results can be more convincing towards his conclusion? Pause video while you think about that and press play when you're ready to continue.

One, the problems here with Jun's experiment is that it only includes 10 trials.

10 trials might not be very many to convince us that one of these outcomes is more likely than the other.

So what we could do is do an experiment with a lot more trials.

Jun conducts an experiment where a frisbee has flipped 500 times and here are the results were displayed in a bar chart, we can see that it landed on heads 281 times and tails 219 times.

So what conclusion could Jun draw from this experiment? Perhaps pause a video while you think about that now and think about how much more convinced are you this time than previously and then press play when you're ready to continue.

Well, Jun says, "These results would suggest that the frisbee is more likely to land on heads than tails." He says, "The frisbee could still land one way more often than the other by chance.

So I cannot be absolutely certain.

However, a conclusion based on 500 trials is more convincing than a conclusion based only on 10 trials." So in one of Jun's previous experiments we did see the frisbee landed on heads more than tails, but that only had 10 trials and they only landed on heads two more times than tails.

That could easily happen by chance.

But this time with 500 trials, the frisbee landed on heads 62 times more than it has on tails and that seems just a little bit more convincing that it is more likely to land on heads and tails.

If you have access to a frisbee and you want to, you could run this experiment yourself by flipping the frisbee in the air a lot of times.

You might not get to 500 but then you could record results and see what you find.

Or if you have access to this slide deck, you can click on the link here and that'll take you to a simulation of this experiment where you can click on run and let the experiment run for itself and see how the results develop.

It should look something a bit like this.

When you open up the Desmos file, you'll find a button in the top left corner, it says run you press pressed out to start and stop the simulation.

Near where it says count results, you'll see a number in a box.

Currently it says zero.

That keeps account of how many trials have been so far in the experiment.

That number will change once we start the simulation and on the right hand side of the screen you'll see a bar chart form showing the number of times each outcome has occurred.

Let's start this simulation and see what happens.

If we stop it here, we can see that currently there's been an equal number of heads and tails, so you might conclude that heads and tails are equally likely to happen.

However, this is only based on 34 trials, the experiment.

So let's see what happens as we continue to experiment with a larger number of trials.

We've now had 400 trials and it looks like it's landed on heads much more than it's landed on tails.

Now, based on these results, we may be confident that the frisbee is more likely to land on heads and tails for two reasons.

One is the number of trials and experiments.

The large number of trials makes it a more convincing argument, and the other is a difference between the number of outcomes that are heads and number of outcomes that are tails.

The greater that difference is for a large number of trials, the more confident you might be, but your conclusion is correct.

So in Jun's experiment where you flip the frisbee air, it landed on heads more times than it landed on tails and Jun has acknowledged that this could just be by chance and actually those outcomes could be equally likely.

So for a comparison, Jun conducts an experiment where a fair coin is flipped 500 times to see what happens there.

Now you may expect when you flip a coin 500 times that you'd get 250 heads and 250 tails because they're both equally likely to happen.

But what we can see here in Jun's results is that's not actually what's happened.

It has landed on heads more times than it's landed on tails, even though they are equally likely to happen.

Jun says, "The frequencies of heads and tails for the coin are not equal, but they are closer than they are for the frisbee.

If you compare these two bar charts side by side, you can see that yes, in both cases you get more heads and tails, but for the coin, which is fair, those results are much closer together.

Whereas for the frisbee, they are much further apart.

So the things from Jun's experiments that might convince us that the frisbee is more likely to land on heads than tails is one the sheer number of trials he's done and if he does more trials that might convince us even more, but also the difference between the heads and tails in that experiment and how much more that is compared to with the- Now, if you want to, you could run an experiment with the coin by flipping it a load at times.

Once again, you might not get to 200, but you could record results and compare it to the results from the frisbee.

Or if you have access to the slide deck here, you can click on this link which shows the results from several different simulations of an experiment with a coin, and you drag the slider to see the results from each different experiment.

It looks something a bit like this.

When you open up this Desmos file, it shows you a bar chart and that is the results from a simulation of an experiment involving flipping a fair coin.

Those two probabilities are set to be exactly equal to each other.

And what you can see is it's landed on tails slightly more, that it's landed on heads.

The slide in the top left, as you drag that, it'll show the results from over experiments based on the exact same probabilities.

And what you can see as you move across them is it's very rare that the coin will land on heads and tails the exact same number of times even though those probabilities are exactly equal to each other.

So let's check what we've learned.

Here we've got three bar charts that show the results from three experiments and each one can see that outcome B happened more, than outcome A.

But which of these sets of results shows most convincingly that outcome B is more likely to happen than outcome A? Pause video while you choose and press play when you're ready for an answer.

The answer is C, and that's because the experiment has the most number of trials.

It's got 1000 trials on experiment.

Okay, it's all due now for task A.

This task includes two questions and here is question one.

Pause video while you do this and press play <v ->When you're ready for question two.

</v> And here are parts A to C for question two, pause video while you do this and press play when you're ready for part D.

And here is part D of question two, which you can do if you have access to the link on this slide and a web browser.

It gives you opportunity to run a simulation of this experiment for yourself.

Use this simulation to conduct your own experiment with a large number of trials.

It's up to you how many trials you do, but try and make it large and then present your results either in a table or a graph.

And you can either do that by hand on paper or you can use some software to help you such as a spreadsheet.

And then based on your results, write down a conclusion about which outcomes you think are more likely to happen than others.

Pause video while you do this and press play when you're ready for some answers.

Let's go through some answers.

Question one had a scenario of dropping a cone and it could land either base down or curve down, and you have the results from two experiments.

Bar chart A would suggest that the cone is more likely to land base down, whereas bar chart B would suggest that the cone is more likely to land curve down.

And then in part C, you have to decide which of those bar charts supports a more reliable conclusion and why.

Well, it'll be bar chart B because that experiment is based on a much larger number of trials.

If you've written something like that, perfect.

You may have also included some additional details.

For example, your answers could also include statements such as we can still not be absolutely certain based on results in bar chart B.

And you may have also said that using an even larger number of trials could produce an even more reliable conclusion.

The scenario in question two related to dropping a toy brick and the different ways it can land, and you've got the results from an experiment with 12 trials Based on the results, it would seem that outcome C is the most likely one to happen, and outcome B is the least likely one to happen.

However, this only has 12 trials, so the experiment could be improved by including a much greater number of trials.

And then if you did part D that allowed you to do precisely that.

You could run the experiment and see what the results were for yourself.

Now when it comes to checking out answers, everyone's answers will be different because it'll depend on how many trials you use in the experiment.

And even people who use the same number of trials are probably gonna have different results.

It depends on the results of the simulation.

Let's take a look, an example answer.

Here's an example answer based on an experiment with 1000 trials.

We can see we have a bar chart of results.

You may have used a table and here's our conclusion on the right hand side, but the order of likelihood would be based on these results that C is the most likely to happen, and then B, and then D, and then A.

However, I look at caveats included here that is if you look at the bars for A and D, yes, D is a little bit higher than A but not very much.

So this answer includes a caveat that D and A may be too close to be conclusive.

Well done so far.

Let's now move on to our next part of this lesson, which is finding and using experimental probabilities.

Previously in the lesson we said that an experiment is a repetition of a trial multiple times in order to observe how often each outcome occurs.

Now the frequency of an outcome from an experiment can be used to produce an experimental probability, and this is sometimes referred to as relative frequency.

Once we have an experimental probability, we may expect that a future experiment would produce the same outcome for a similar proportion of its trials.

We might not necessarily expect the exact same proportion of trials to be the same outcome, but a similar one.

And even then this might not necessarily happen.

Probabilities don't tell us exactly what's going to happen, they just give us an informed guess for what could happen.

However, the more trials that an experiment contains, the more confident that we can be about any experimental probabilities that are inferred from it.

Let's look at an example now.

Here we have Jacob who conducts an experiment where a frisbee has flipped 500 times and the bar chart shows results.

It shows that the frisbee landed heads more often than it landed tails.

Therefore, if Jacob did another experiment, he could expect that it would land on heads more than tails in that experiment as well.

But what if Jacob wanted to try and predict how many times it would land on heads in a future experiment? Well, that's where an experimental probability would come in.

The results from an experiment can be used to produce experimental probabilities by dividing the frequency of each outcome by the total number of trials.

For example, to get the probability that the frisbee lands on heads, we could do 281 divided by 500.

281 is a number at times it landed on heads in this experiment, and 500 was the number of trials in our experiment, and that will give a probability of 0.

562, or it could be just expressed as a fraction or even as a percentage.

That's fine as well.

The probability of getting tails would be 219 over 500, which is no 0.

438.

Jacob says, "I wonder what results I would get if I repeated the same experiment." Do you think you would get exactly 281 heads and 219 tails again? Well, let's take a look.

Here are are the results from another experiment where a frisbee has flipped 500 times.

The results are shown in this bar chart.

Based on these results, what is the experimental probability that the frisbee lands on heads? And give your answer this time as a fraction.

Pause video while you do it and press play when you're ready for an answer.

The answer is 275/500 or you can use any equivalent fraction to that.

What is the experimental probability that the frisbee lands tails? And this time, give your answer as a decimal pause video while you do that and press play when you're ready for an answer.

The answer is 0.

45, which you get from dividing 225 by 500.

Here are the results from another experiment.

An experiment is conducted where each trial has three possible outcomes, A, B, and C, and the table showed results.

Let's start by asking how many trials were conducted in total during this experiment? Pause video while you write it down and press play for an answer.

The answer is the sum of these three frequencies and that is 500.

So if there were 500 trials altogether based on these results, what is the relative frequency or experimental probability for outcome A? Pause video while I write it down and press play when you're ready for an answer.

The experimental probability for outcome A could be expressed as either a fraction, decimal, or percentage.

Either as 81/500 or 0.

162 or as 16.

2% or anything equivalent.

Let's now go back to Jacob and his experiments involving in flipping a frisbee.

The experimental probabilities for each of Jacob's two experiments with the frisbee can be seen on the screen.

Now, what these probabilities show us is the proportion of times that the frisbee landed on each outcome during those experiments.

And what we can see is they weren't exactly the same, but they were pretty similar.

0.

562 is pretty close to 0.

55.

Jacob says, "The results were not exactly the same, but they were similar.

So if I conducted other experiments, I could expect results to be in a similar proportion again.

Again, let's take this experimental probability here.

The probability of getting heads is 0.

55.

That is the proportion of times it landed on heads in a previous experiment.

So let's now use that to make some predictions about another experiment.

Based on the experimental probability we can see here, approximately how many times could Jacob expect to get ahead in 1000 trials? Now, this probability was based on 500 trials, and what we're trying to use it for is predict the outcomes for 1000 trials.

So even though the number of trials is different, we would still expect that a similar proportion of the outcomes would be heads.

So we could calculate this in a couple of different ways.

One way could be to use equivalent fractions.

On the left, we have the fraction that is the probability of getting heads 275 over 500, and what we want is a fraction that is equivalent to that with a denominator that is 1,000, and that would be 550/1,000.

Therefore, we could conclude that out of 1,000 trials, we would expect around 550 of them to result in heads.

That might not necessarily happen, but it's a good approximation based on the data that we have.

An alternative method could be to use a decimal.

We could do 0.

55 multiplied by 1,000, and that again would give 550 and lead us to the same conclusion.

So based on the experimental probability, Jacob would expect 1000 trials to result in approximately 550 heads.

Jacob conducts the experiment and the bar chart shows the actual results for an experiment with 1000 trials.

What do you think about those results? How do they compare to what we expected? Pause video while you think about it and press play when you're ready to hear from Jacob.

Jacob says, "There wasn't exactly 550 heads, but 539 is pretty close to what I expected." So probabilities can't tell us exactly what is going to happen, but it can give us a good sense of what we might expect based on previous data.

If you'd like to, you could click on the link on this slide to take you to a frisbee simulation on Desmos and run this experiment several times.

You could run it once for a large number of trials, work out an experimental probability based on your results, predict how many times it would happen again in another experiment, and then run the experiment and see if the results are what you expected.

So let's check what we've learned.

Here we have an experimental probability of getting heads as 0.

55.

Based on this experimental probability, approximately how many times could the outcome heads be expected in 100 trials? Pause video while you work this out and press play when you're ready for an answer.

The answer is 55 times, and there are two methods on the screen to show how you can get that.

How about if we only express this probability as a decimal? How many times could you expect the outcome heads to occur in 850 trials? Pause video while you do this and press play when you're ready for an answer.

Well, if you multiply 0.

55 by 850, you do not get a whole number.

You get 467.

5.

Now it can't land on heads 467.

5 times, so you'd predict it'll land on heads either 467 times or 468 times.

Okay, it's all due now for task B, this task contains two questions, and here are parts A to C of question one, pause video while you do this and press play when you're ready for part D.

And here is part D, which you can do if you have access to the link on this slide and a web browser.

You can use that link to open up a simulation of this experiment, press the run button to start and stop it.

And what your task is here is to run your simulation 400 times and then compare your results to what you expected from your answers for part B and part C.

Pause video while you do this and press play when you're ready for question two.

And here is question two, pause video while you do it and press play for some answers.

Okay, let's go through some answers.

In question one, we had the scenario of a cone being dropped and it could land either base down or curve down.

We have an experiment with 100 trials and the results meet are shown in the bar chart.

In part A, you have to work out the experimental probability or relative frequency of each outcome.

You should have 0.

32 and 0.

68, or you can express that as a fraction or a percentage.

And then based on results, you have to say, which outcome would you expect to have the highest frequency in a future experiment? Well, it would be outcome B because it would seem based on the previous experiment that that outcome is the most likely to happen each time.

And then part C, based on these results, approximately how many times would you expect each outcome to occur in 400 trials? You could do that by multiplying each experimental probability by the number of trials.

0.

32 times 400 is 128.

0.

68 times 400 is 272.

So we'd expect it to land on A and B that many times each, but it might not necessarily do that, and that's what Part D was about.

You could run the experiment yourself or a simulation of the experiment and see how many times it landed on A, and B.

Now if you did so, answers may vary depending on the simulation results, but here's an example answer for what you could write for part D.

You could write something like the frequency of each outcome was not exactly the same as what was expected in Part C.

It might have been, but possibly not.

However, what I suspect a lot of you'll have noticed is that outcome B still happened more often than outcome A, and that was predicted in part B.

So as discussed earlier, probabilities don't tell you of absolute certainty what is going to happen in any situation, but what they can do is give you a good reliable approximation for what you might expect based on previous data.

Then question two involves a scenario about a toy brick being dropped and you got the results from an experiment in the table.

In part A, you had to write down the relative frequency of each outcome or the experimental probability.

You would do that by first working out the total frequency and then you could produce these relative frequencies.

In Part B, you had to approximate how many times you could expect outcome C to occur in 80 trials.

You'd do that by multiplying experimental probability by 80 to get a 30.

8.

Now it can land that way 30.

8 times, so you'd expect either 30 times or 31 times.

And then part C, you had a chance to run the simulation for yourself for 80 trials and compare your results to part B.

Here are some example answers for what you could put.

'cause once again, your results may vary depending on the simulation.

One example answer is that the frequency of outcome C was not exactly the same as what was expected in part B, but it was close.

Or you may have said something like the frequency of outcome C was quite different to what expected from Part B, but outcome C still occurred the most often, or you may have written something different that was relevant for your results.

Fantastic work today.

Now let's summarise what we've learned.

A set of results can be critically evaluated to help determine if a game or an object is fair.

Different samples or experiment sizes may affect your perception of fair and the experimental probability of an outcome or event can be calculated from a set of results by taking the frequency of that event or outcome and dividing by the total frequency in the experiment.

Experimental probabilities can be used to estimate the number of times an event may occur in a set number of trials.

Well done today.

Have a great day.