Lesson video

Hi, my name's Mrs. Wheelhouse.

Welcome to today's lesson on using two-way tables to display outcomes for two events.

This lesson falls within the unit on Probability, Possible Outcomes.

I love probability.

So I'm really looking forward to doing today's lesson with you.

Let's get started.

By the end of today's lesson, you'll be able to systematically find all the possible outcomes for two events using a two-way table.

Let's get started.

So a key phrase that we're using today is outcome table.

Now, an outcome table is a table that shows all the possible outcomes.

So really, it's just a way of displaying information.

Let's see how we're going to use it today.

Our lesson today has three parts.

And we're gonna begin by looking at an outcome table from a two-stage trial.

If we were to spin this spinner twice, what are the possible outcomes? Well, we can make an outcome table to determine possible outcomes.

The columns will be spin one.

And the rows refer to spin two.

The number of columns will be the number of possible outcomes from the first spin.

And the number of rows are the number of possible outcomes from the second spin.

We can then fill in the spaces with the possible outcomes.

So if I win on my first spin and win on my second spin, then I can see that the outcome will be win-win.

I could also lose-win, win-lose, and lose-lose.

So an outcome table will help us determine the possible outcomes.

On this spinner, you can see we have three possible outcomes on a spin, which is to win, lose, or draw.

And so, my outcome table has gotten larger.

As you can see, I can still win-win, win-lose, lose-win, lose-lose.

But now, I have to consider what happens if this spinner shows draw.

Let's do a quick check.

I'd like you to pause the video to complete this outcome table.

Do that now.

Welcome back.

Let's check to see if you filled in the table correctly.

So you should have, if I read the first row, AA, BA, CA, second row, AB, BB, CB, and then the third row, AC, BC, CC.

Now remember, it's okay if you have those two letters in the different order.

So you may have written, for example, instead of BA, AB.

And then in the other position, written BA.

That's fine if you have.

Let us consider this particular situation.

So a trial is to spin both of those spinners.

We could make an outcome table to show the possible outcomes.

So spin one is in my columns and spinner two is my rows.

As you can see, I filled in all the possible outcomes by considering for each cell what's the column that it's in and what row it's in.

Sam says, "Does it matter which spinner was spinner one?" Well, in this case, no, because getting 1A and A1, they are the same outcome.

Which of the following could be an outcome table for these two spinners? So consider the possible outcomes for each.

Which table do you think represents an outcome table for these two? Pause the video where you make your choice.

Welcome back.

What did you put? Well, it should have been both A and B.

For A, you can see that we've put the second spinner, so the one with the letters on, as the column headings.

Whereas B, it's for the rows.

And that's all we've done.

It's the same table just swapped around.

But for C on the other hand, I've been mixing up which outcomes go on which position.

So C is definitely not right.

For example, it's not possible for me to spin both a one and a two by spinning those two spinners, once each.

A game is being played where the highest spin wins.

Sophia and Jun are playing the game.

And they both have a copy of the spinner.

And you can see we've drawn a table here to represent the results.

Now, both of these outcomes are where one spinner has a one and the other spinner shows a two.

So for example, if Sophia spins and gets a two and Jun spins and gets a one, then Sophia wins because she has the higher number.

However, the allocation of the number to a person is now really important because it determines who wins.

So unlike before, when 1A and A1 were the same, this time around it really matters who spun and got what number.

A game is being played where the total from two spins of this spinner determines the number of counters collected.

An outcome table can show the possible totals, which allows you to see the likelihood of receiving that number of counters.

So for example, if I spin that spinner twice and get a five each time, then I'll receive 10 counters.

So the number of possible counters I might receive after spinning twice goes from two all the way up to 10.

Now, it has to be said.

These outcomes are not equally likely to each other.

Can you see why? That's right.

You can see that there's only one 10 in the table.

Whereas, the outcome of getting seven appears four times.

Quick check now.

The two spinners are going to be spun.

And the numbers will be multiplied together.

Please pause the video now to complete the outcome table.

Welcome back.

Check your table now.

Well done if you've got these all right.

A trial consists of spinning these three different spinners.

Can an outcome table be constructed in this situation to show the possible outcomes from this trial? What do you think? In order to do this, we'd need a three-dimensional table.

Now, it's gonna be really difficult to see every outcome, and in fact, really hard to draw this on a page.

Another way though would be to complete a two-way outcome table for two of the spinners, and then use those outcomes with the last spinner.

So for example, let's look at the spinners that show one, two, three, and the ones that show A, B, C, D.

I've completed my outcome table.

And then what I'm going to do is combine those.

So it's like I'm imagining there's a brand new spinner that comprises the two previous spinners.

So effectively, I've put them together.

And on it, I can see all the outcomes that appeared in this table.

So what I'm now gonna do is consider those two spinners, so my new combined one and my last original one.

So you can see across the top, so the headings of all the columns are from my new spinner.

And then the heart and the club that are on my second spinner make up my rows.

Now, I can see all the outcomes and these are all the outcomes from spinning my three spinners.

So it is possible to have done this.

But it is a lot of work.

Quick check, which of the two spinners have been combined to make this outcome table? Welcome back.

Did you go for A and C? You can see that the spinner that has AB on it is the one that wasn't combined.

And the spinner that has one, two, three and shows heart, club, those two were combined because they made our new column headings.

It's now time for your first task.

In question one, we're going to spin this spinner twice as part of a two-stage trial.

Please complete the outcome table to show all the possible outcomes.

Pause and do this now.

In question two, both of these spinners are going to be spun as part of a two-stage trial.

Complete the outcome table to show all the possible outcomes.

Pause while you do this.

Welcome back.

Question three, Laura and Izzy are playing a game of rock, paper, scissors.

Just in case you don't remember the rules, rock beats scissors, scissors beats paper, and paper beats rock.

Please complete the outcome table to show when each person will win.

Make sure you are really clear in each cell as to who is winning.

Pause and do this now.

Question one, you can see the outcome table showing all the possible outcomes here.

Do feel free to pause if you need to to check your answers.

Question two, here's our completed outcome table.

Again, feel free to pause to check your answers.

Now for three, you can see here that I filled in the table stating who wins in each situation.

And for some of them, it's a draw.

So if you wrote draw instead of no one, that's absolutely fine too.

It's now time for the second part of today's lesson.

And we are going to be looking at outcome tables from two events.

If this spinner is spun, the possible outcomes are one, two, three, four, five.

Now, Izzy and Lucas are playing a game.

And Izzy gets a point if it lands on a prime number.

And Lucas gets a point if it lands on an odd number.

Now, an outcome table can be set up to categorise the possible outcomes.

So you can see across the top we've got when Izzy gets points and when she doesn't.

And when Lucas gets points and when he doesn't are our rows.

All possible outcomes, or our sample space, are represented in the outcome table.

So we can see one, two, three, four, five has been placed in our table.

And they're placed when they meet the conditions of both events.

So for example, three is an odd number.

But it's also a prime.

And that's why it's going in that top cell.

Let's check we've got that.

Izzy and Lucas are deciding to change the rules of their game.

Izzy now gets a point if the spinner lands on an even number.

And Lucas gets a point if the spinner lands on a factor of six.

Please pause and complete the outcome table.

So put in the outcomes in the places where they meet the conditions.

Pause and do this now.

Welcome back.

Let's see if you put these in the right places.

So one is not an even number, but it is a factor of six.

So you can see where we place one.

Two is both even and a factor of six.

So it's gone in that top cell.

Three is not even, but it is a factor of six.

So it's joined to the one.

Four is even, but it is not a factor of six.

And five is not an even number nor a factor of six.

So it's placed in that very last cell.

This outcome table has been created for two events from a trial.

What is the sample space for this trial? How am I going to work that out? Well, actually it's pretty straightforward.

The sample space for this trial is simply the outcomes I can see in my table.

So that's the numbers one, two, three, four, five, six, seven, eight, nine.

What are the two events I'm considering? Now, you might think there's more than two events.

But actually, if you look very carefully, there are only two events I'm interested in.

Really good.

Well done if you spotted that.

The first event is I'm considering if a number is odd.

And then the second event is considering the square numbers.

Now, it's absolutely fine if you wrote these the other way around.

And then we can write down which outcomes fall into each event.

Which outcomes can you see that are in both events? What have you spotted? It's one and nine.

We can see them both in the set of outcomes for event one and in the set of outcomes for event two.

Quick check now.

Use this outcome table to write down the sample space.

Pause the video while you do this.

Welcome back.

You should have written down one, two, three, four, five, six, seven, eight, nine, because they are all the outcomes that appear in our table.

Time for our second task.

This spinner is going to be spun.

The two events that we're considering are, is the number even? And is the number a factor of 14? Complete the outcome table to show the outcomes for each event.

Pause and do this now.

Welcome back.

Question two, please sort these drinks into the outcome table.

So you're considering, does the drink have a straw? And does the drink have a handle? Pause and do this now.

Welcome back.

Let's mark question one and question two.

So for question one, this is where you should have placed those outcomes.

Feel free to pause the video while you check your work.

And now, question two.

This is where you should have sorted the drinks.

Again, feel free to pause while you check your work.

It's now time for the final part of today's lesson.

And here, we're going to be looking at frequency in a two-way table.

Here are some party hats from a pack.

Now, some of them can be said to be striped.

And some of them can be said to have toppers.

So remember, striped ones could be anything from straight stripes to curvy stripes to wavy stripes.

And toppers refer to the thing at the top of the hat.

So has it got something sitting on the top of it? So we can sort them into an outcome table based on whether or not they have something on the top and whether or not they could be called stripy.

So this is where I'm placing the first hat.

It's not striped but it does have something on the top.

And then my second hat, this is striped but doesn't have something resting on the top.

Let's place the rest of the hats.

There we go.

So you can see how we're categorising them.

Rather than have the individual party hats though, we could just focus on how many there are, so the frequency.

There are two party hats which are both striped and have toppers.

So we could just say there are two that fit that category.

And there are four which have a topper but are not striped, and three that have stripes but no toppers, and then just one that has no stripes and no topper.

Rather than have the individual party hats, we could now just focus on the frequency.

So we could actually work out the total that was striped and not striped, and the totals that had toppers and no toppers.

So there were five striped party hats in total and five that were not.

There are six with toppers and four without.

We can clearly see here that there are 10 in total.

So five out of five, or six out of four.

Another pack of party hats has been sorted by whether they're striped and/or have toppers.

And the results are shown in this two-way table.

How many striped party hats are in the pack? Pause while you work this out.

Welcome back, now remember, I asked for how many striped party hats there were.

I didn't say whether or not it needed to have a topper.

So you have to say that's not a concern.

It's just how many are striped.

So you should have said there are six in total.

How many party hats didn't have a topper, so no topper in the pack? Pause while you write your answer.

So you should have said there were four that didn't have a topper.

How many of the striped party hats didn't have a topper? Pause while you work this out.

So this time, we were being more restrictive.

It had to be a striped party hat with no topper.

So that's just two.

Alex is running a charity raffle.

The tickets are either yellow, blue or pink.

And each has a three digit number on.

He decides to consider the two events.

The ticket is yellow and the ticket has an odd number.

He has a partially completed two-way table here.

Do you think there's enough information for him to complete it? What do you think? I think there's definitely enough information.

The first thing I can do is fill in the total column.

So 20 and 19 is going to give me 39.

The next thing I'm going to do is I'm gonna fill in that first row.

I know that there are 20 odd tickets in total, of which six are yellow.

So that must mean that 14 are not yellow.

I'm gonna fill in the next row.

If I know there are 19 even tickets, of which 13 are not yellow, then six must be yellow.

And now, I can fill in both of those columns.

I know there's a total number of yellow tickets that are 12.

And there must be 27 that are not yellow.

And I can check that I've done this right because 12 at 27 does indeed make 39.

So it's a great thing to do here.

What I've done is just to go back and just check.

The order which this two-way table was completed could be different.

So I could have done it in a different order.

As long as you know two values in either a row or a column, you're able to calculate the missing third value.

And it's really important at the end that you can always use the final value to act as a self-check by calculating it from both the row and the column.

Let's do a quick check now.

Here's another partially completed two-way table regarding shoe size and year group in school.

Which of the entries can be calculated first? Remember, there may be more than one.

Pause the video while you work out what can be calculated.

Welcome back.

Which ones did you go for? You should have said we can calculate A, the number of year nine students with not size seven feet.

Because if we look at the not size seven column, I know two of the values in it and I'm only missing one, which is the top one.

So I can definitely work that out.

And the other one I could have calculated is the number of not year nine students who have size seven feet.

If I look at the not year nine row, I know two of the values in it.

And I'm only missing one.

So I can calculate that missing one.

It's time for our final task.

For question one, a random sample of people who's being selected by allocating an integer from one to 50 to each person, and then picking based on two rules.

Rule one is that the integer is a multiple of 10.

And rule two is the integer is a multiple of four.

Please complete the two-way table to show the frequency for each thing happening.

Pause and do this now.

In question two, we have 10 pupils that have been asked about whether or not they're members of a sports or music club.

So for part A, circle all the numbers in the table that include Izzy.

And then for B, which clubs does Sophia do, if she does any? Pause while you do this.

Question three, fill in the missing boxes on each of the two-way tables.

Pause and do this now.

Welcome back.

Let's go through the answers.

So for question one, asked you to complete this two-way table.

And you had to work out, remember, how many values fitted into each category? Well, there are only two values that are both multiples of 10 and multiples of four.

And they would be the numbers 20 and 40.

They are both multiples of 10 and multiples of four.

And they fit in the range one to 50.

So that's why I wrote two in that very first cell.

I've then gone through and checked where each number falls and completed my table by counting up how many outcomes fall into each space.

Feel free to pause while you check your table against mine.

Welcome back.

Question two asked you to circle the numbers on the table that include Izzy.

Now, we can see here that Izzy had nothing filled in.

So Izzy does not do a music club.

And she does not do a sports club.

That means that the number one in the no music club and no sports club cell has to refer to Izzy.

And that means she's being counted in both the total of the students that don't go to music club, and the total for the students that don't do a sports club.

This means she's counted in the overall total as well.

So which clubs does Sophia do, if any? Well, by working out where everybody else goes, we can see we had a spare person that did music but did not do sports.

And that must be where Sophia is.

Question three asked you to fill in the missing boxes on each of the two-way tables.

So you should have filled them in as follows.

Do feel free to pause while you check your answers.

It's time to sum up our learning today.

Outcomes can be listed in an outcome table.

And for a two-stage trial, one stage will be the columns and the other one will be the rows.

The sample space is the outcome shown within the grid.

It's a systematic way to construct a sample space.

For two events from a trial, the sample space can be categorised by the events.

When frequencies are shown in a two-way table, it is possible to find missing values using the totals for the columns and the rows.

Well done.

You've done a great job today using two-way tables to display outcomes for two events.

I hope you enjoyed the lesson and found it interesting.

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