Lesson video

Hello, I'm Mrs. Lashley and I'm really looking forward to working with you throughout this lesson today.

So I'm hoping you're ready to make a start.

In today's lesson, we're gonna be working out the theoretical probabilities for combined events from different forms of two-way tables.

On the screen you can see the definition for theoretical probability.

You may wish to read through it just to familiarise yourself, make sure you're happy with that as we get going.

So our lesson's got two learning cycles.

The first part is thinking about theoretical probabilities.

Once again, always from outcome tables.

And then the second part is probabilities for combined events.

So we're gonna make a start looking at theoretical probabilities from two-way tables or outcome tables and the different forms of those.

So on the screen we've got a deck of cards that are, the cards numbered one to 10 and we can sort them into odds and evens and we can do that using a table.

So if we've got one column that represents the cards that are odd numbers and one column that represents the numbers that are even, and that would be one, three, five, seven, and nine.

And the remainder are the even numbers, two, four, six, eight, and 10.

We could also sort the same 10 cards into square numbers, so those that are square numbers and those that are not square numbers, once again using a table is a nice way of doing that organising.

So the square numbers one, four and nine, one squared is one, two squared is four, three squared is nine, and therefore the rest of them are not square numbers.

And we've got two separate tables there, one that's showing the odds against the evens and one that's showing the square numbers against the numbers that are not square.

But we could combine these two tables into what's commonly known as a two-way table or an outcome table.

And so here we can see the same tables that we had previously, so the columns is square one, four and nine, and they're not square, the rest of them.

And the rows are the odd numbers, one, three, five, seven, and nine, and the even numbers two, four, six, eight, and 10.

So both of the tables we had sorted them for those two different events and we've now combined them.

So if a card was gonna be chosen randomly from this deck, what's the probability that you would get an odd number? Five out of 10, there are five possible desired outcomes, one, three, five, seven, and nine out of the 10 in the sample space.

So what's the probability of selecting a square number? Three out of 10.

If you selected card one, card four or card nine, then you would've got a square number.

So there were three desired outcomes, three possible ones that you would want out of the 10 in the sample space.

So the probability is three out of 10.

This is a theoretical probability, don't forget, it may not work out that if you did 10 selections, 10 random selections, that you would get a square number three of the times.

What's the probability that you would get a card that is both odd and square? So just think about that one.

So you've got 10 cards, if you randomly select one, what would be the probability in theory that you should select a card that is both odd and square? Two out of 10, and this is where the two-way table, the outcome table is really useful.

We can see the combination of going across the row, which is the odd row, so all of those cards on that row represent odd numbers and down the square column, and therefore the cell where they intersect are the cards that are both odd and square or both even and square or both odd and not square.

There are four sort of possibilities.

So a different scenario here.

We've got drinks that have been sorted into an outcome table according to whether the drinks are in a cup that has a handle or whether the drink has a straw.

Those drinks have been assigned a letter so that we can sort of sort them into that outcome table.

So again, if a drink was to be chosen at random, with no influence, you couldn't see, you're just blindfolded or something, what's the probabilities that a randomly selected drink would have no handle but would have a straw? So no handle and a straw.

So whereabouts on the outcome table are the drinks that have been sorted into that category of not having a handle but do have a straw? Well, it's that cell, so it's there's three of them.

If you look at the images, D is in a glass without a handle but does have a straw, E, glass without handle but does have a straw, and H glass without a handle, does have a straw.

So there are three possible desired outcomes out of the nine drinks in total.

We could simplify that fraction to a third, give it as a percentage, give it as a decimal, but I'm just gonna leave it as three out of nine.

What's the probability that you would randomly select a drink that has a handle but doesn't have a straw? So you could do that with the images, but if I've removed the images on the outcome table, where are you gonna find the outcomes that have a handle but no straw? Well, it's only one outcome that fits that description, and that is the drink A.

So if you look at drink A in the image, you can see it looks like a mug of tea perhaps.

So it's got the handle and you're not drinking it through a straw.

And what's the probability if you randomly selected a drink that you would get one with a handle and a straw? There are four that would be okay for that one, and that is in that part of the outcome table, four out of nine because there are nine in the sample space, there are nine possible drinks that you could select.

So a check for you, this spinner is to be spun just the once and what is the probability that it lands on a number that is odd and a factor of 10? So pause the video whilst you decide on the probability, and when you're ready to check, press play.

So you are looking for the probability that it lands on a number that is both odd and a factor of 10.

That would be if it landed on number one or number five.

And so, that's two possible outcomes out of the sample space of five.

So two out five, which is equivalent to 40% or 0.

4 Another check, what's the probability that it lands on a number that is even and not a factor of 10? Pause the video whilst you decide on that probability and then press play to check it.

So even and not a factor would be the number four, that would be the outcome on the spinner that you're looking for.

So there is only one desired outcome out of the five possible on the spinner.

So the probability is one fifth, which is 20% or 0.

2.

Here we've got a game that's going to be played where the players spin the spinner twice and find the sum of their numbers.

So that spinner has got five possible outcomes, one, two, three, four, and five.

They're gonna spin it twice and find the sum of their numbers.

So the outcome table shows you the possible totals.

So the columns represent what you could get on your spin one.

So you could get one, you could get two, you could get three, you could get four, you could get five.

The rows represent the outcomes that you could get on spin two.

But our sample space is about the possible totals because you're gonna sum them.

So you can see that if you got one on your first spin and three on your second spin, then your sum is four.

Event A we are gonna say is even outcomes.

So even sums and they've all now got a border with sort of a dashed border in purple.

Event B are outcomes that are more than five.

And so, they've got the green dashed border.

So if you were looking for the probability that when the player spins the spinner twice and finds the sum of their numbers, their sum would be even and more than five.

So it would be A and B, it's these outcomes on the outcome table that have both the purple and the green because they are both outcomes for being even and outcomes for being more than five.

So what's that probability? Well, there are nine if you count how many of those outcomes there are, nine out of 25.

There are 25 possible sums, not unique some, some of the totals come up more than others, but there are 25 possible sort of combinations.

And so, that would be our probability.

So the outcome table shows you all of those possible sums from the two spins.

And now we can make a two-way table for the two events that we defined.

We defined the event A to be even and we defined the event B to be more than five.

So we can go from our outcome table to a two-way table like we've seen previously.

So the total and the total total was 25.

There are 25 possible outcomes.

If we just focus on the outcomes that are even, there are 13 of those.

So the row for A, the row for event A, there is a total of 13.

and that leaves 12 that are not in event A.

Because event A is even, not being even makes you an odd number.

So there are 13 even totals and there are 12 odd totals.

If we then look at the event B, which is outcomes where they are more than five, there are 15 of those and you can see them all highlighted there on the outcome table.

Well, if there are 15 outcomes that are more than five, then that means that there are 10 that are not more than five, there are 10 that are five or less.

And you can see they are the outcomes that have not got the green border.

And now if we focus into the more combination part of the two-way table, so there are nine outcomes that are both A and B, there are nine outcomes that are even and more than five and they're there on the outcome table.

And we use that for our probability, which means that there are four that are even numbers but are not more than five because that gives us the total of 13.

We can also use that nine and the total of 15 to calculate that there would be six outcomes that are odd but more than five.

And you can see them that they're the four sevens and the two nines, they are more than five but they are odd numbers, not even.

And then lastly, there's a few ways you could figure out the missing number in that cell, it will be six, six because six add six gives you the total of 12 for the not A, and six because four add six gives you the total of 10 for not B, also nine add four, add six plus six would give you the total of 25.

So now that we've sorted our outcome table into a two-way table, this will help us to work out probabilities more efficiently.

What is the probability that the sum is not even and more than five? So which part of that two-way table would allow you to get the probability in a really efficient way? So not even, remember that event A is for being even, and more than five, which is event B.

Well, it's the six that are not even and more than five out of the 25.

25 possible outcomes, that's the sample space, and six of them are the desired outcome that we're looking for, which is not even and more than five.

What's the probability that the sum is not even and not more than five? Well, this is also six out of 25 because there are six outcomes that are not even, so not A, and six outcomes that are not more than five.

Here's a check.

Again, you've got a given outcome two-way table here based on two events and the two events are about party hats.

And the party hats have been sorted into whether they are striped, the stripes could be vertical, horizontal, zigzagged like you can see in the examples of some party hats, and whether they have a topper.

So if a party hat is chosen at random, what is the probability that it is not striped and has a topper? So pause a video whilst you're using the two-way table to decide on your probability.

So if a party hat is chosen at random, what is the probability that it is not striped and has a topper? Press play when you want to check your answer.

You got five out of 13.

So that's because there are five hats that are not striped with a topper and there are 13 in total.

So five desired out of the 13 in the possible sample space.

And the sample space is the pack of party hats.

Same context, same outcome table, but this time the question is if a party hat is chosen at random, what is the probability that is striped and has a topper? So once again, pause whilst you are taking the numbers from that outcome table to get your theoretical probability and then press play when you're ready to check.

So this is four out of 13.

The sample space is still 13, it's still the amount of hats that are in a pack, but there are four of them that we could define to be stripy and with a topper.

So we're up to the first task of the lesson for learning cycle one.

And so here you need to part A is fill out the outcome table, and then part B, C and D is working out probabilities based from your outcome table.

So pause the video whilst you're doing question one, and then when you press play, we'll go to question two.

Question two, again, part A is about you completing a outcome table showing the possible products.

So this time the spinner is to be spun twice, but the product of the two numbers is to be found.

And then parts B and C is to use your outcome table to work out some probabilities.

Press pause whilst you're working through the question two.

And when you press play, you've got one more question of this task.

Here's question three, so question three, this two-way table shows the frequencies of some sports balls categorised by their shape and whether they have stitching on the surface.

So for example, an American football is not spherical but does have stitching, it's got like the lacing on there, whereas a golf ball is spherical but doesn't have stitching.

So a ball is to be selected at random and you've got two probability questions to find.

So find the probability that it would be a spherical ball that has stitching and part B is find the probability that it would be a spherical ball but does not have stitching.

Press pause whilst you're working those two probabilities out.

When you press play, we're gonna go through the answers to this task.

So question one's on the screen.

Firstly, you needed to sort the outcomes from the spinner into the outcome table.

So prime numbers and not prime numbers, and then whether they were less than 12 or whether they were not less than 12.

And then part B, C and D was working out probabilities.

I've given the probabilities as all the different forms that you could have possibly given.

So fraction, simplified fraction, decimal or percentage, you just needed to have one of those written down.

But it's worth you noticing and reminding yourself about the equivalent forms. Question two, the spinner was to be spun twice and a product of the numbers was going to be found.

So product means to multiply together.

So first of all, for part A, you needed to complete all of the possible outcomes.

There were 25 possible outcomes.

And then for part B and C, you needed to use that to find the probabilities.

So for part B it was find the probability that the product would be square and a factor of 36 and there were five that were square and a factor of 36.

And that is number one, so the outcome of one because one is square, one squared and also one is a factor of 36.

Four, there were three fours that you could have got and then also nine, and there was only one way that you could get nine and that was by getting three on both spins.

So that was five in total and there are 25 possible outcomes.

Once again, the equivalent forms are given, you needed to only to have one of those.

Part C, find the probability it's even and less than 18.

So going through row by row, if it's even and less than 18, count it.

And there were 14 of those, and that's 14 out of 25, which is equivalent to 0.

56 and 56%.

Question three, so about the sports balls and whether they were spherical or not and whether they had stitching or not, the outcome table was there, you needed to just use it to get your probabilities.

So the probability of it being spherical and with stitching, there were four types of sports ball that could be classified like that out of the 18 in total, so four out of 18, that does simplify to two ninths, which is 0.

2 recurring or 22.

2% recurring.

So because of the recurring nature of that decimal, it's probably more useful to leave it as a fraction so it doesn't get any rounding error.

And then part B, find the probability its spherical and does not have stitching, well, there were 11 sports balls that come under that classification out of the 18.

So 11 out of 18.

We're now up to this lesson's second learning cycle where we're going to finding probabilities for combined events.

We've done a little bit of that in the learning cycle one, but we're gonna take it a little bit further in this second learning cycle.

So here we've got a spinner with outcomes one to five.

We are assuming that the spinner where all the outcomes are equally likely to come up and we have then got an outcome table where we've categorised two events.

One event is whether it's a factor of 10 and the other event of whether it's even, so you've got factor of 10 or not a factor of 10, and even or not even, but obviously if it's not even we say odd.

So you can see the outcomes have been sorted.

So what is the probability that the outcome is even or odd? So if you was to spin that spinner, what's the probability that you will land on an outcome that is even or odd? Well, that's any of the outcomes because if you land on one, that's odd, if you land on two, it's even.

If you land on three, it's odd, if you land on four, it's even, if you land on five, it's odd.

So what's the chance, what's the probability that you land on a number that is even or odd? Well, all of it, so five out of five, all five of the outcomes are even or odd.

And you can see that on the outcome.

And so, that is a probability of one.

It is certain that we would land on a number that is either even or odd.

What's the probability that the outcome is even and a factor of 10 or odd and not a factor of 10? Two combinations combined together here.

Even and a factor of 10.

Well, we can use the outcome table to see that that's if you land on number two.

So if you land on number two, that's an even and a factor of 10.

And then it's or landing on odd and not a factor of 10.

Again, if we use the outcome table, you can see that the outcome of three is odd and not a factor of 10.

So there are two outcomes that would satisfy this combination out of the five possible outcomes that you could land on on that spinner.

So two out of five.

What's the probability that the outcome is even and not a factor of 10 or odd? So that first part of the question is even and not a factor of 10.

Going to the outcome table even and not a factor of 10 is the outcome four.

And then it's all landing on an odd number.

Well, where on the outcome table are the odd numbers? Well, it's the whole bottom row.

So that means that there are four outcomes that satisfy this combined event.

Four out of five, which is equivalent to 80% or 0.

8.

So here's a slightly different spinner and a slightly different outcome for your check.

What is the probability that the outcome is odd and a factor of eight or even? So use the outcome table to identify which outcomes fit this, how many there are out of the total possible sample space.

Pause the video whilst you're doing that and then press play to check your answer.

That's four out of six, which is equivalent to two thirds.

Another check, what is the probability that the outcome is odd or even and not a factor of eight? So once again, use that outcome table to identify the outcomes that would satisfy this.

Press pause whilst you have a go.

And then when you're ready to check press play.

So it is the whole bottom row 'cause it was being odd.

So if you spun the spinner and you landed on an odd, that was fine, or if you landed on a number that was even and not a factor of eight, and the only outcome that satisfies that is six.

So there was four different outcomes that you could have landed on where you would be odd or even and not a factor of eight, four out of six, which is equivalent to two thirds.

So continuing with these combination of events, we've got Lucas and Sam playing a game with this spinner, and Lucas is gonna get a point if its spinner lands on a factor of 10, Sam gets a point if the spinner lands on an even number.

So this outcome table, instead of putting the events, we've put whether Lucas gets a point or whether he doesn't get a point.

So you can see that from the outcomes of the spinner.

So what's the probability that Lucas and Sam get a point? Well, that only happens if they land on two.

When they spin the spinner, if it lands on number two, both of them will get a point because two is a factor of 10 and it's also an even number.

So the probability that they both get a point is one sixth.

What's the probability that Lucas or Sam get a point? So Lucas could get a point and Sam doesn't, and that would be if we land on a one or a five, Sam could get a point and Lucas doesn't, and that's if the spinner was to land on four or six, but they could also both get a point and that would be if the spinner landed on number two.

So there are five outcomes on that spinner that means that somebody gets a point.

It doesn't matter if it's Lucas or if it's Sam or if it's both of them, as long as somebody is getting a point.

So here is a check, which of the following satisfies an outcome being even or a factor 50? So read through those four and decide which of them satisfies an outcome, be it even or a factor of 50.

Press pause whilst you're doing that and then when you're ready to check, press play.

Well, A, B and D would satisfy that.

So if the outcome is even and not a factor of 50, then that's okay because we were looking for it being an even or a factor of 50.

So if it's even then we're happy.

For part B, the outcome is even and is a factor of 50.

Once again, we wanted something what was even or a factor of 50.

Well, if it's both, then that's great, it satisfies both of the requirements.

And lastly, D, the outcome is odd, which means it's not even and is a factor of 50.

So that one doesn't satisfy being even, but it does satisfy being factor of 50.

So that's also okay.

So we've got a spinner that's gonna be spun and then we're gonna find the sum of their numbers, we've seen this previously and we've got our outcome table that shows us the number of outcomes in each combination, whether it's in A and whether it's in B, whether it's not in A, whether it's not in B, whether it's in A and B, et cetera.

So what is the probability that the sum is even or more than five? This part of the two-way table represents the number of outcomes that are in event A and event A is about being even.

So we are looking for the probability that the sum is even or more than five.

So if you got any of those 13 sums, you would've satisfied it being even.

And this part of the two-way table represents those that are more than five, and there are 15 of those that come out as more than five, but our probability would only be 19 and that's because there are nine that satisfy both A and B.

So if you go back to that check and Lucas and Sam's playing that game, if we're looking for them being even or more than five, there are three ways that would satisfy that.

And the three ways are it being an even number and not more than five, and that is four on this outcome table, or it could be odd and more than five and that's the six.

Or it could be both even or more than five and that is nine.

So the combination there of even or more than five means that they come together as 19 possible sums out of the 25.

What is the probability that the sum is even and not more than five or more than five? So what is the probability that the sum is even and not more than five? So first of all, let's answer that bit.

Where on the two-way table will we find the number of outcomes for that? And then we'll look at numbers or outcomes that are more than five.

So being even, which is event A, and not more than five, which is event B, is there are four possible sums that give you that outcome.

And then where on this is it more than five? Well, more than five is event B, so it's nine and six.

And so this has the same total, it's the same outcomes that are be included here.

So here's a check for you.

An eight sided dice has the numbers one to eight.

What is the probability that when it is rolled it lands on a factor of eight, which we're calling F, or an odd number, which we are calling O? So use the two-way table to help you with this.

Identify which outcomes for each part of the combination and then find the total.

Pause the video whilst you're doing that and then when you're ready to check it, press play.

So we have a total of seven outcomes that satisfy being a factor of eight or being an odd number, and there are eight in total, so seven out of eight.

So we're onto task B for this lesson.

Question one, you've got an outcome table that shows you the sample space of the integers from one through to nine and an outcome is going to be randomly selected and you need to find the probability according to those four questions.

Press pause and then when you are ready, we're gonna go through the answers as there's only one question to this task.

So probability there's even and a factor of 18 or odd is seven out of nine.

So you would be including all of the odd outcomes, which there are five.

And you're also looking for any outcomes that are even and a factor of 18.

And there were two of those, two and six.

So five plus the two gives you seven outcomes that are desired, are the ones that satisfy that statement out of the nine possible ones.

Part B, odd and a factor of 18 or even.

So odd and a factor of 18, there are three of those, one, three and nine, and an even there are four outcomes.

So in total that is seven.

So there are seven desired outcomes out of the nine in the sample space.

Part C, a factor of 18 or even and not a factor of 18.

So we're gonna break it into the two sort of parts to this or statement.

So a factor of 18, well, there are five of those, two, six, one, three and nine, and then even and not a factor of 18, there are two of those, four and eight, there's no overlap between those two.

So we are gonna total when it comes to seven out of nine.

And then lastly, D, not a factor 18 or an even outcome.

So again, sort of breaking it into the two parts of the or statement.

So how many outcomes are not a factor of 18? Well, there are four of them, which is four, eight, five and seven.

And then how many outcomes are even? Well, there are four of those, which is two, six, four, and eight.

But four and eight are outcomes that are both even and not factors of 18.

So I've included them twice, so we don't want to include them twice.

So it would be two, six, four, eight, five, and seven are the outcomes that I want to count, which means there are six of them out of nine.

That does simplify to two thirds.

So if you did simplify your fraction to two thirds, that's great, well done, but six out of nine.

So it comes to the end of the lesson.

So to summarise, outcome tables can be used to find theoretical probabilities of combined events.

Really useful if you've got a physical one to be able to sort of identify which outcomes you actually want and then you can count them up to see which ones are the desired outcomes.

But importantly, and what's also useful about being able to do it physically is that in some combinations an outcome will appear in both and therefore you only include it once.

So like we just saw on that task, that if the outcomes are in both of the events, you don't want to count it twice.

So being able to just identify the outcomes that satisfy both parts of the combined event is really useful.

Really well done today and I look forward to working with you again in the future.