Lesson video

Hello there, and thank you for joining me.

My name is Dr.

Rowlandson, and I'll 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 versus theoretical probability.

And by the end of today's lesson, we will be able to see the impact of sample size and compare theoretical probabilities to experimental ones.

Here are some previous keywords that will be useful during today's lesson, so 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, and we're going to start with using and interpreting experiments.

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

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

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

Now, these two outcomes are equally likely to happen, and that's because of the shape and weight of the coin.

The shape is pretty much symmetrical between heads and tails, and the weight is pretty much uniformly distributed between heads and tails as well.

So that makes it equally likely to land on either side.

Because these two outcomes are equally likely to happen, it means we can find a theoretical probability of each outcome.

The probabilities should sum to one, and they're equally likely, so you could do one divided by two to get a probability of one half for each of those outcomes.

Now, can you think of any other examples of situations where theoretical probabilities can be found? Thinking particularly about the shape and weight of objects, which means that the outcomes are equally likely to happen.

Perhaps pause the video while you think about this and press play when you're ready to continue together.

Let's take a look at some examples then.

When rolling a fair coin, each outcome is equally likely to happen.

Again, that's because the shape of the dice and the weight of the dice means it's equally likely to land on each side.

So we can find theoretical probabilities of each outcome.

There are six different outcomes, which are all equally likely.

The probability should sum to one.

So if we do one divided by six, we get a probability of 1/6 of it landing on each of its faces, and that's a theoretical probability.

Another example is this spinner here.

Now, the outcomes for this spinner are not equally likely.

You can see that it's more likely to land on C than it is on A.

However, the spinner is equally likely to land on each sector.

All those sectors are the same size.

So because it's equally likely to land on each of its individual sectors, it means we can calculate theoretical probabilities for this spinner.

We could do one divided by six, because there are six sectors, and then multiply it by the number of sectors for each letter.

Those probabilities would be a probability of 1/6 for A, 2/6 for B, and 3/6 for C.

So theoretical probabilities can be found in cases where either events are equally likely to happen or events can be considered as groups of outcomes, which are all each individually equally likely to happen.

This is often the case when using objects that are symmetrical, have sides or faces of equal size, or have uniform weight throughout.

For example, flipping a coin, rolling a dice, or spinning a spinner with equally sized sectors.

In cases where differences of size or mass within an object may affect the likelihood of an event, it may not be possible to find theoretical probabilities.

Let's take a look at some examples.

When flipping a frisbee, it could land either one side or on the other, but the weight is not equally distributed between those two sides, so it may be more likely to land on one side than the other.

That means we can't calculate a theoretical probability for that particular event.

When dropping a toy brick such as the one you can see here, it can land on different sides, but those sides are not equally likely and it can't be split up into smaller outcomes that are equally likely either.

So a theoretical probability cannot be calculated there either.

And the same as well when dropping a cone.

It may be more likely to land one way than another, so once again, a theoretical probability cannot be calculated by simply dividing one by a number and then multiplying it by some other number.

So if we can't calculate theoretical probabilities for these situations, what can we do? Experimental probabilities may be derived instead by collecting data, and this is also true in cases where differences of skill, strength, experience, knowledge, or other conditions may affect your likelihood of an event.

It may not be possible to find theoretical probabilities in these situations either.

Let's take a look at some examples of that.

In football, you usually have two teams playing against each other.

Those two teams are not necessarily equally likely to win.

It might be that one team is better than the other team when it comes to skill or experience or whatever.

Therefore, a theoretical probability cannot be calculated in that situation.

Another example is when athletes run a race.

Let's say for example, you've got seven athletes who are all about to run a race.

It wouldn't necessarily be correct to divide one by seven to say that the probability of each person winning is 1/7, because some of those runners might be faster runners than others.

Some might be more knowledgeable about how the conditions might help them or maybe have more experience or so on.

There's lots of different things that may affect their probabilities.

So we can't just calculate a theoretical probability by dividing one by seven.

Another example is two players playing chess.

One of those players may be more knowledgeable and experienced than the other player, and therefore may be more likely to win than the other person.

So if we can't get the theoretical probabilities for these situations, what could we do? Experimental probabilities may be derived by analysing previous data.

How have all these people performed in previous events? Let's take a look at a practical example, which you could replicate yourself if you want to.

Here we have Andeep who has a box containing 10 counters.

All the counters are either blue, green, or red.

We have Laura and Sam as well.

They do not know how many counters there are of each colour in the box.

Andeep says, "I've got a challenge for you.

Can you estimate how many counters there are of each colour in the box?" He says, "You are not allowed to look in the box, and you are only allowed to take one counter out at a time." Hmm, how could Laura and Sam produce a reliable estimate for how many counters that there are of each colour in the box? And the key there is the word reliable, not just guess any old number.

How could you have a reliable estimate? Pause video while you think about this and press play when you're ready to see what they do.

Let's see what they do.

Laura and Sam plan an experiment to produce a reliable estimate for how many counters there are of each colour in the box.

Laura says, "We could take a counter out of the box, record its colour, place it back into the box, and then repeat the process." Sam says, "We could use the results to estimate how many counters of each colour are in the box." So that's what they do.

They conduct an experiment with 10 trials.

They chose 10 trials because there are 10 counters in the box.

And the table shows the results.

They drew a blue counter out the box five times, a green counter out the box twice, and a red counter out the box three times.

And remember, they always put the counter back in the box before they do it again.

They use their results to estimate how many counters of each colour are in the box.

And Sam says, "We think there are five blue counters, two green counters, and three red counters in the box." Hmm, why might the conclusion be incorrect? Pause the video while you think about this and press play when you're ready to continue.

Well, it seems their conclusion is based on the idea that if there are 10 counters in the box, and they do 10 trials, then they are likely to draw each counter out exactly once, but that's not necessarily true.

They may not have necessarily drawn each counter out of the box exactly once.

They may have drawn some counters multiple times and not drawn other counters times at all.

So how could they improve their experiment? What could they do differently to produce a result that is more reliable as an estimate than what they've done so far? Pause the video while you think about this and press play when you're ready to continue.

Laura says, "We could use a greater number of trials.

But then that would lead to numbers that do not add up to 10, and we know there are 10 counters in the box.

So what could we do then? Sam says, "We could use the results to produce experimental probabilities or drawing each colour, which could inform our estimate.

So that's what they do.

Here we have now a table of results for an experiment with 50 trials.

We can see they got blue 16 times, green eight times, and red 26 times.

They use the results to produce experimental probabilities of drawing each colour.

The probability then based on these results of drawing blue would be 16/50, which is 0.

32.

The probability of drawing green will be 8/50, which is 0.

16.

And the probability of drawing red will be 26/50, which is 0.

52.

What these experimental probability show us is the proportion of times that each colour was drawn during that experiment.

So they could assume that the number of counters of each colour in the box are in the same proportions or similar proportions to what they drew.

How could they produce an even more reliable experimental probability for each colour? What could they do? Pause the video while you think about this and press play when you're ready to continue.

Well, they could use an even greater number of trials.

So over to you then.

Here we have a table that shows results for an experiment with 200 trials.

Could you please use the results to write down the experimental probability of drawing each colour? Pause the video while you do this and press play when you're ready for some answers.

Here are the answers.

We take the frequency for each colours drawn and divide by the total frequency for the experiment.

So for blue, 24 over 200 will be 0.

37.

For green, 18 over 200 will be 0.

9.

And for red, 108 over 200 will be 0.

54.

You can express your answers as the fractions, as the decimals, or as percentages or any equivalent fractions as well.

So let's now go back to the point where they had their experiment with 50 trials.

The table shows the frequency that each colour was drawn from the box, and also the experimental probability of drawing each colour based on those results.

Now we know that there are 10 counters in the box in total.

They use their experimental probabilities to estimate the number of counters of each colour in the box.

Let's see how they do it.

Sam says, "We could assume that our results are in the same proportion to the colours inside the box." Now, it's worth stressing that it's an assumption that those proportions are the same, but it's the best reliable guess we can use based on the data we have so far.

So that's what we're going to do.

We're going to estimate the number of counters in the box by multiplying the experimental probability for each colour by 10, because there are 10 counters in the box altogether.

So for blue, we do 0.

32 multiplied by 10 is 3.

2.

Now, we can't have 3.

2 blue counters in the box.

There must be an integer number of counters, so we'd round it to the nearest integer, which is three.

For green, we do 0.

16 multiplied by 10 is 1.

6.

So we could estimate that there are two green counters in the box.

And for red, we could do 0.

52 multiplied by 10 to get 5.

2, and estimate that there are five counters in the box.

And it's worth double checking that those numbers add up to the total numbers of counters in the box.

Three plus two plus five is 10.

So those are our estimates based on those experimental probabilities.

Over to you now to do a similar thing.

Here we have a table that shows the results from an experiment with 200 trials.

You've got the frequency for each one.

We've also got the experimental probabilities of drawing each colour based on those results.

You're told that there are 10 counters in the box.

Use the experimental probabilities to estimate how many counters there are of each colour in the box.

Pause while you do it and press play for answers.

Here are the answers.

If you multiply each experimental probability by 10 and then round to your nearest integer, you should get four blue counters, one green counter, and five red counters.

Now, you may notice that those numbers are slightly different to the estimates we had in the previous experiment.

Let's compare 'em side by side.

Here we have the results from two experiments, experiment A and experiment B.

And you can see that the estimated number of counters is slightly different for each.

Which experiment is the most likely to have produced a reliable conclusion? And please explain why.

Pause the video while you do that and press play for answers.

Experiment B will be the most likely to have produced a more reliable conclusion because it used a greater number of trials.

Now that doesn't necessarily mean that the conclusion from experiment B is exactly correct, but it's likely to be more reliable than the results from experiment A.

So here's the moment you've been waiting for, and it reveals the 10 counters that are in the box.

He says there are four blue, one green, and five red counters.

Based on the actual number of counters in the box, what are the theoretical probabilities of drawing each colour from the box? Pause the video while you write them down and press play when you're ready for answers.

If we assume that each individual counter is equally likely to be drawn from the box, then we could do one divided by 10 to find the probability of drawing each individual counter and then multiply it by the number of counters there are for each colour.

So we'd get these.

For blue, it'd be 4/10 or 0.

4.

For green, it'd be 1/10 or 0.

1.

And for red, it'd be 5/10 or 0.

5.

Based on the theoretical probabilities, approximately how many times would you expect to draw each colour in an experiment with 200 trials? Pause the video while you write these down and press play when you're ready for answers.

We'd expect the number of outcomes for each colour in 200 trials to be in the same proportion as their theoretical probabilities.

So if you multiply each theoretical probability by 200, you would expect to draw blue 80 times, green 20 times, and red 100 times.

Now, that might not necessarily be what exactly happens, but based on the probabilities, that is a reliable estimation for what could happen.

For our final question together, let's compare the experimental probabilities from three experiments with the actual theoretical probability.

In the table, you have the experimental probabilities based on 10 trials in the column labelled A, you've got the experimental probabilities based on 50 trials in the column labelled B, and from 200 trials in the column labelled C.

On the left, you have the theoretical probabilities based on the actual number of counters in the box.

Which set of experimental probabilities are the most similar to the theoretical ones? Is it A, B, or C? Pause while you choose and press play when you're ready for answers.

The answer is C.

The experiment based on 200 trials seems to have produced probabilities that are the most similar to those which are their theoretical probabilities.

Now, they're not exactly the same, but they are the closest out of all the experiments.

So in this practical scenario, we've been able to calculate experimental probabilities based on the results from experiments, but we've also been able to see what the theoretical probabilities are based on the actual number of counters in the box.

And by doing so, we've been able to observe that the greater the number of trials in the experiment, the closer the experimental probabilities became to the theoretical ones.

So we may conclude that the greater number of trials you have in an experiment, the more reliable your experimental probabilities are likely to be.

Okay, it's over to you now for task A.

This task has one question and it's broken into four parts.

Here is part A of question one.

Pause the video while you do it and press play when you're ready for part B.

And here is part B.

Pause while you do it and press play for part C.

Here is part C.

Pause while you do it and press play 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.

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

You press the run button to start and stop the simulation.

And could you please, if you do this, use this simulation to conduct an experiment of your own with a large number of trials, and then present your results either in a table or on a graph.

You could do that on paper if you want to, or you could use a software such as a spreadsheet software to do so.

Then based on your results, estimate the number of counters of each colour in the box.

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

Okay, let's go through some answers.

You have this scenario where you've got a box containing 20 counters, which are all either blue, red, or green.

Alex plans an experiment where each trial consists of drawing a counter from the box, recalling its colour, and placing it back in the box.

The table in part A shows the results from 10 trials, and you have to produce experimental probabilities and then estimate the number of counters of each colour.

The experimental probabilities would be 0.

4 for blue, 0.

5 for green, and 0.

1 for red.

Or you can use equivalent fractions or percentages.

And if you use those, you would estimate the number of counters of being eight for blue, 10 for green, and two for red.

Part B was the same thing, but based on 400 trials.

And then when you calculate the experimental probabilities, you should get 0.

495 for blue, 0.

29 for green, and 0.

215 for red.

And that would give you estimated numbers of 10 blue counters, six green counters, and four red counters.

Part C, you have to decide which set of results would produce a more reliable estimate for the number of counters in the box.

That would be the results from experiment in part B, because they're based on a much larger number of trials.

And then part D, you've got a chance to have a go at it yourself by running a simulation.

Now the results for this will be different depending on the number of trials you ran and also what the results from your simulation was.

'Cause each time you run it, it's a little bit different.

Here's an example of an answer about using an experiment with 2,000 trials.

The results are presented in a bar chart and also in a table as well.

We can see the frequency for each one, which you may have to zoom in quite a bit to read off the bar chart what those frequencies are.

We've got estimated probabilities, which are all multiplied by 20, to give estimate number of the counters as nine for blue, seven for green, and four for red.

You may have the same numbers, you may have different numbers.

But if your numbers are different, I wonder if you still have blue as being the highest number and red as being the lowest number.

You're doing great so far.

Now let's move on to the next part of this lesson, which is conducting experiments to test fairness.

Here we have Alex and Izzy who are playing a game that involves flipping a coin.

The bar chart shows the frequency of each outcome after 10 flips, and what we can see is it's landed on heads seven times and tails three times.

Wonder what Alex and Izzy have to say about this.

Alex says, "I think the coin is biassed, because the coin showed heads more often than tails." Where Izzy said, "I think the coins are actually fair, but it just showed heads more than tails by chance." What do you think? Who do you agree with here? And how could they find out? Pause the video while you think about this and press play when you're ready to continue.

In theory, when outcomes have equal probabilities, we could expect them to occur the same number of times.

However, in practise, this does not necessarily happen.

One outcome may occur more often than another simply by chance.

And this is particularly the case when an experiment contains only a small number of trials.

Therefore, when one outcome occurs more often than another, it does not necessarily mean that the trial is biassed or that the same outcome has a greater probability of happening in the future.

You could try this out for yourself.

You could flip a coin 10 times and see if you get five heads and five tails.

You might do, but you might not do.

And I suspect if you did get one outcome more often than another, you probably wouldn't be surprised by that.

The same as well when you roll a dice.

You don't always necessarily roll each number an equal number of times.

Sometimes you may roll one number more often than another, and that's just by chance.

It doesn't necessarily mean that the dice is biassed.

We can see an example of what we're talking about here with these four bar charts.

They each show the results from an experiment where a coin is flipped 10 times.

And you can see in some experiments, it landed tails more than heads.

In other experiments, it landed heads more than tails.

Each set of results can be used to produce experimental probabilities, and here they are.

The probability getting heads in the first experiment will be 0.

2, in the second will be 0.

6, in the third will be 0.

8, and in the fourth will be 0.

3.

If you have access to this slide deck and a web browser, you could click on that link and it'll take you to a Desmos file with even more simulations of flipping a coin.

You drag the slider across to view the results from each different simulation.

Alex looks at these results and he says, "Sometimes there are more heads, but other times there are more tails." And Izzy says, "The experimental probability of getting heads varies quite a lot between experiments." Definitely does, and that's because it's based on a very small number of trials, only 10 trials.

Let's see what happens when we do an experiment with many more trials.

These four bar charts show the results from experiments where a coin is flipped 100 times.

Now, once again, we can see that sometimes it lands on heads more than tails, and sometimes it lands on tails more than heads.

The results vary between.

Each set of results can be used to produce experimental probabilities, and we can see here we have 0.

49 for heads in one experiment, and the next experiment, it was 0.

59, and then 0.

47, and 0.

42.

If you have access to a slide deck and have a web browser, you could click on the link to take you to some Desmos simulations of the same thing with 100 trials and see how other results vary as well.

Alex makes the same observation.

He says, "There are still sometimes more heads, and other times more tails." But Izzy says, "The experimental probabilities don't vary as much as they did in the experiments with fewer trials." Yes, these probabilities are all different, but they're all much closer together than they were when we only had 10 trials.

In fact, these probabilities seem to be getting pretty close now to 0.

5.

So we've seen that the more trials we have, the closer the experimental probabilities get to being the theoretical probabilities.

I wonder what happens if we have even more trials.

Here we have four bar charts that show the results from experiments where a coin is flipped 1,000 times.

And once again you can click on the link to access more experiments with 1,000 trials.

Each set results can be used to produce experimental probabilities again, which you can now see underneath the bar charts again.

And Alex says, "The frequency of each outcome still isn't equal in any of the experiments." But what we can see here is the experimental probabilities are now very, very close to the theoretical probability of getting heads with a fair coin.

They are all so close to 0.

5.

So based on these experiments, it would seem like the coin is not biassed, it's fair, because the experimental probability is very, very similar to the theoretical probability.

Let's think about what we can take away from this.

Experiments can be used to investigate whether the outcomes of a trial are equally likely and generate experimental probabilities.

This can be useful for checking whether a game is fair or biassed or any other situation.

But remember, even if the outcomes are equally likely, for example, if a game is fair, then the frequencies of each outcome may still differ.

So just because one thing happens more often than another, it doesn't necessarily mean that it's always more likely to happen.

However, the greater number of trials, the more we may expect that the proportion results for a particular outcome to be similar to its actual probability.

A larger number of trials may be needed to be confident about the conclusions from an experiment then.

So let's check what we've learned.

A spinner contains three letters, A, B, and C.

The table shows results from two experiments without spinner.

Which experiment would produce the more reliable set of experimental probabilities and explain why.

Pause while you do it and press play for answers.

The answer would be experiment two, because there are much greater number of trials in that experiment.

So let's use experiment two.

Could you please calculate the relative frequency, or experimental probability, in other words, for each outcome? Pause while you do it and press play for answers.

Well, the total frequency is 2,000.

So if you divide the frequency of each outcome by 2,000, you'd get these numbers on the screen.

If the spinner was biassed, which outcome would seem the most likely? Pause the video while you write that down and press play when you're ready for an answer.

If the spinner was biassed, then you would expect outcome A to be the most likely as it has the highest relative frequency.

True or false? Based on the results, we can be absolutely certain that the spinner is biassed towards A.

Is that true, or is that false? And write down a reason why.

Pause the video while you do that and press play for answers.

The answer is false, and the reason why is even with a large number of trials, results can differ between experiments due to chance.

Therefore, we cannot be absolutely certain based on these results.

Okay, it's over to you now for task B.

This task contains two questions, and here are parts A and B of question one.

Pause video while you do it and press play for the rest of question one.

And here are parts C and D to question one.

Pause while you do it and press play for question two.

Here are parts A and B of question two.

Pause while you do it and press play for the rest of question two.

And here are parts C and D of question two.

Pause while you do it and press play for some answers.

Okay, let's go through some answers.

In question one, we had the situation where a frisbee is flipped.

In part A, you could say that the probability of getting heads is 0.

5.

In part B, you could say the probability in heads is 0.

545.

In part C, you have to say which probability would seem the most reliable.

That would be your probability from part B, because it's based on an experiment with a greater number of trials.

And then for Part D, you had to decide whether you thought the frisbee was fair or biassed towards a particular outcome, and justify your answer.

Well, the results from experiment in part B would suggest that the frisbee is slightly biassed towards heads.

However, you cannot be absolutely certain.

Then in question two, you had this scenario where a class was trying to work out whether a dice was biassed.

They conduct an experiment in part A, and you have to get the relative frequencies or the experimental probabilities.

They will be these numbers here.

Then part B, you had to do the same again, but with a different experiment, and that will produce these relative frequencies here.

In Part C, you have to decide which of those experimental probabilities were the most reliable.

That'll be the experimental probabilities from part B, because they were based on much greater number of trials.

And Part D, you had to decide whether you thought the dice was fair or biassed and justify your answer.

Well, the results from experiment in part B would suggest that the dice is fair, because the experimental probabilities are very similar.

They're not exactly the same, but they're pretty close to each other.

However, we cannot be absolutely certain.

Fantastic work today.

Now let's summarise what we've learned.

It may not always be possible to calculate theoretical probabilities, so in situations where you can't, relative frequencies can be collected from data to produce experimental probabilities.

It's worth bearing in mind that different sample sizes or numbers of trials can impact these experimental probabilities.

As a sample size increases, the experimental probabilities approach the theoretical probabilities, if there are some.

And a larger sample size is more likely to show a true estimate of the underlying probability.

Well done today, have a great day.