Lesson video

Hello there and welcome to today's lesson.

My name is Dr.

Rollinson and I'll be guiding you through it.

Let's get started.

Welcome to today's lesson from the unit of probability and theoretical probabilities.

This lesson is called Calculating Theoretical Probabilities from a table where we have one event, and by the end of today's lesson, we'll be able to find theoretical probabilities from a table showing the possible outcomes for one event.

Here are some previous keywords that we're gonna use again during today's lesson.

So you may wanna pause the video if you want to remind yourselves about the meanings of any of these words.

Then press play when you're ready to continue.

This lesson contains two learn cycles where we're gonna look at finding probabilities from a couple of different types of tables.

In the first learn cycle, we're gonna focus on frequency tables.

Then the second learn cycle, we'll go look at outcome tables, but let's start off with finding probabilities from a frequency table.

Here we have Alex.

Alex has a box of novelty flavoured sweets.

Some of the flavours taste nice, but some of the flavours have been manufactured to taste nasty as a joke.

The flavours of the sweets are shown in a table.

So in the left-hand column we have the tasty flavours, strawberry, caramel, toffee, fudge, orange, and chocolate.

Now, you might not necessarily think that all those flavours are tasty, but those are the flavours that the manufacturers have labelled as tasty ones.

And then the second column has the nasty flavours in this box, they are grass flavour, earwax flavour, fish and garlic flavour.

Now you might think that some of those flavours sound tasty, but those are the ones that the manufacturers have labelled as nasty flavoured sweets.

Alex puts one of each flavour into a bag and then he offers a sweet to Izzy.

He's feeling a little bit mischievous today.

What is the probability that Izzy picks a nasty flavour from the bag? Well, let's think about this.

There are 10 sweets in the bag and four of those sweets are nasty flavours.

So if we write the probability that Izzy chooses a nasty flavour as a fraction, we would have 4/10.

Where four represents the number of sweets in the bag that are nasty flavours, and the 10 represents a total number of sweets in the pack.

So here Alex has put one of each flavour sweet in the bag.

But let's now take a look at the entire box of sweets.

The table shows how many of each flavour sweet are in the entire box.

On the left-hand column, we have the flavours that are available, and on the right-hand column we have the frequency of each flavour.

For example, the row it says strawberry says frequency of 11.

That means that 11 of the sweets in that box are strawberry flavour.

On the bottom row says total, that says 80.

That means there's 80 sweets in the box altogether.

So what is the probability of randomly choosing a fudge flavour sweet from the box? Let's think about this together.

There are 12 fudge flavoured sweets in the box because the frequency of fudge is 12 and there are 80 sweets in the box altogether because the total frequency is 80.

So if we write this probability as a fraction we'd have 12/80.

Where the numerator here represents the number of fudge sweets in the box, and the denominator represents the number of sweets altogether in the box.

This probability could also be written as a decimal, as 0.

15 as a percentage, 15%, or as a simplified fraction as 3/20.

It could also be written as any equivalent fraction as well.

So what is the probability of randomly choosing an orange flavour sweet from this box? Let's do this one together.

What is the frequency of the orange flavour sweets? That would be eight, and the total frequency is 80.

So the probability of choosing an orange flavour sweet from the box can be written as a fraction as 8/80, where eight represents the number of orange sweets, and 80 represents the number of sweets altogether in the box.

Now Izzy's two favourite flavours are fudge and orange.

So what is the probability that she randomly chooses one of these flavours? Well, let's think about how many sweets in that box are one of her favourite flavours.

We have 12 fudge sweets and we have eight orange sweets.

So that means there are 20 sweets in that box, which are one of Izzy's favourite flavours.

So the probability is 20/80 where 20 comes from the sum of the fudge and orange frequencies, and the 80 is a total number of sweets in the box.

So let's check what we've learned.

This table shows information about pupils on a school trip.

We have Year 7, 8, 9, and 10 and 11, and then we have the frequency of each year group.

So there are eight pupils on the trip from Year 7 and so on.

A pupil is chosen at random for a task while they're on the trip.

What is the probability that the pupil chosen is from Year 10? Pause the video while you write down the probability and then press play when you're ready for an answer.

The probability of randomly choosing a Year 10 for this would be 7/40, where seven represents the frequency of the Year 10s and 40 is the total frequency, the total number of pupils on the trip.

You could also write this as an equivalent fraction or an equivalent decimal if you want to as well.

What is the probability that the pupil chosen is from Year 7 to 9? Pause the video while you have a go at this and press play when you're ready for an answer.

The answer is 24/40.

The 24 comes from the sum of the frequencies for Year 7, 8 and 9.

So that's eight plus six plus 10 makes 24, and the 40 is a total frequency.

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

This task contains two questions, and here is question one.

Here once again, we have a box of novelty flavoured sweets.

The table shows a number of sweets of each flavour in this novelty box of sweets.

You need to first, find the probabilities for picking each of the following flavours when a sweet is chosen at random and the flavours are there.

And then part B says, the nasty flavours are the grass, earwax, fish and garlic sweets.

What is the probability of choosing a nasty flavoured sweet? Pause the video while you have a go at this then press play when you're ready for question two.

And here is question two.

Sam is carrying out a data investigation about how pupils in a nearby village travel to school.

She conducts a survey and records her results in a tally chart.

The first thing you need to do is complete the frequency column of that tally chart, and then you have some questions about the probabilities for each mode of transport.

Pause the video while you have a go at this and press play when you're ready to go through some answers.

Right, let's see how we got on with that then.

In question one, we defined the probabilities of picking each flavour of sweet from the box.

So the probability of choosing a strawberry flavour sweet is 11/80, caramel is 10/80, toffee is 8/80, chocolate 9/80, grass is 8/80 as well.

Ear wax is 3/80, fish is 5/80, and garlic is 6/80.

And then the nasty flavour sweets are the grass, the earwax, the fish, and the garlic.

What is the probability of choose a nasty flavoured sweet? That'll be 22/80.

And then question two, we had to complete the frequency column of the table.

So 20 people travel by car, 12 people travel by bus, nine people by bicycle and 16 people walk to school.

And using this, we then find some probabilities.

So what's the probability that the person chosen travels by car? That would be 20/57, where 20 is a frequency of car and 57 is the total frequency.

The probability of travelling by bus is 12/57, by bicycle is 9/57, and walking is 16/57.

Well done so far.

Now let's move on to the second learn cycle, which is finding probabilities from an outcome table.

Here we have a deck of cards that contain the numbers one to 10.

There's one of each card.

Now with this deck of cards, we could sort them in lots of different ways, but let's start by sorting them into odds and evens using the table below.

Our odd numbered cards are one, three, five, seven, and nine, and our even numbered cards are two, four, six, eight, and 10.

Now that we've rearranged these into a table, it can make it easier to see different probabilities.

For example, if a card is chosen at random from the deck, if we wanna find the probability that a card is odd, then we could look at how many odd numbered cards there are.

There are five cards which are odd, and we can look at how many cards there are altogether.

There are 10 cards altogether.

So this probability as a fraction would be 5/10.

We could simplify it to 1/2 or write it as 0.

5 or 50%.

But 5/10 is a clear way to write this fraction based on this scenario.

And the probability that the card is even is also 5/10 as well.

There are five even cards and there are 10 cards altogether.

We could also sort the cards into primes and not primes using a table as well.

So our cards, which are prime, are two, three, five and seven.

And our cards that are not prime are one, four, six, eight, nine and 10.

Using this, we could find some probabilities.

If a card is chosen at random from the deck, the probability that the card is prime is? Well, there are four cards that are prime and there are 10 cards altogether.

So the probability is 4/10.

And the probability it's not prime? Well, there are six cards that are not prime and there are 10 cards altogether.

So it's 6/10 or any fractions that are equivalent to these or decimals or percentages that are equivalent as well.

Now so far we've used one table to sort into odds and evens and we use a different table to sort into primes and not primes.

We could use a table to sort it both ways, so the carts could be sorted into odds and evens as well as primes and not primes using this table.

The columns show prime and not prime and the rows show odd and even.

So if we sort these cards into this table, looking at the top left empty box, which is prime and odd, those cards would be three, five, and seven.

In the box that is underneath that one where it says prime and even that would be the number two.

And then in the top right-hand box, which is for not prime and odd, those are one and nine.

And then for the bottom right-hand box, which is not prime and even, that's the remainder of the cards, it's four, six, eight, and 10.

Now, using this table, we could find some probabilities again.

A card is chosen at random from the deck, the probability that it's an even prime? Well, let's take a look.

In the box that is for cards that are even and prime there's only one card, the number two, and there are 10 cards altogether.

So the probability of choosing an even prime would be 1/10.

The probability of choosing an odd prime? Well, in the top left-hand box we have the numbers which are odd and prime.

There are three of those and there are 10 numbers altogether.

So the probability of choosing an odd prime is 3/10.

So what about the probability of choosing a number that is not prime and is odd? Well, if we look at the column that says not prime, and we look at the row that says odd, we can see that there are two cards that fit into both of those.

One and nine are not prime and they are odd.

So that's two out of 10, 2/10.

So let's check what we've learned.

The spinner contains the numbers one to five, sort the five outcomes into the table below.

Pause the video while you quickly draw this table and write the numbers from one to five in the right places in the table.

And then press play when you're ready for an answer.

Okay, let's go round the numbers from one to five.

The number one is odd and is a factor of six.

The number two is even and is a factor of six.

Three is odd and is a factor of six, four is even but is not a factor of six and five is odd, and it's not a factor of six.

So our answer should look something like this.

Let's now use this table.

The spinner is spun once.

What is the probability that it lands on an odd factor of six? Pause the video while you write down an answer and press play when you're ready to see what the answer is.

Well, there are two numbers on the spinner that are odd factors of six, that is one and three.

So the probability as a fraction would be 2/5, or you could put it as an equivalent decimal 0.

4, or 40% as a percentage.

What is the probability that it lands on an even factor of six? Pause the video while write down this answer and press play when you're ready to see what it is.

There is one number here that is an even factor of six.

The number is two, and there are five numbers altogether.

So the probability as a fraction will be 1/5.

It'd also be 0.

2 or 20%.

What is the probability that it lands on a factor of six, regardless of whether it's odd or even? Pause the video while you have a go at this and press play when you're ready to continue.

The answer is 3/5.

That's because there are three numbers here that are factors of six.

One of them is even and two of them are odd.

There are three altogether.

So it's over to you now for Task B.

This task contains two questions, and here is question one.

We have a bunch of party hats.

Party hats from our pack are sorted according to how they are decorated, and we can see our outcome tables shows us that.

The columns show us whether or not they have stripes, and the rows show us whether or not they have a topper.

So based on that table, find each of the following probabilities and you've got six different probabilities to find.

Pause the video while you have a go at this question.

Then press play when you're ready for question two.

And here is question two.

A deck of cards contains the numbers one to 20.

And here we have an outcome table where the columns show us whether or not the card contains a multiple of four.

And the rows show us whether or not the card contains a factor of 20.

The first thing you need to do is sort the numbers one to 20 into this outcome table.

And then in part B, you can use that outcome table to find the probabilities in those questions.

Pause the video while you have a go at this and press play when you're ready to go through some answers.

Okay, let's see how we got on with that.

In question one, we need to find some probabilities based on this table about the party hats.

So part A says, what's the probability that a hat chosen at random has stripes and a topper? Well, there are two of those hats and there are 10 hats altogether.

So the probability is 2/10.

In part B, the probability that a hat chosen has stripes and has no topper? Well, there are three hats that fit those two descriptions, and there are 10 hats altogether.

So the probability is 3/10.

And part C, the probability that it does not have any stripes, but it does have a topper will be 4/10.

There are four hats that have no stripes and have a topper, and there are 10 hats altogether.

Part D, no stripes and no topper? That would be 1/10.

Part E stripes regardless of whether or not it's a topper? Well, that'll be 5/10.

And then F, the probability that it has a topper is 6/10.

And then question two, we need to first sort the numbers one to 20 into the table.

Here's where all the numbers go.

You may want to pause the video if you wanna check your table against this one here.

Now let's use this table to find some probabilities.

In question B, part one, the probability of choosing a card that contains a multiple of four and a factor 20 is 2/20.

In part two, the probability of choosing a card that is a multiple of four and is not a factor 20 is 3/20.

The probability of choosing a card that is a multiple of four, regardless of whether or not it is a factor of 20, is 5/20.

And in part four, the probability of choosing a card that is not a multiple of four is 15/20.

Fantastic work today.

Let's now summarise what we've learned in this lesson.

The probability of an outcome can be found by considering a frequency table showing all the possible outcomes.

And in many cases, when we are finding probabilities from a frequency table as a fraction we'll find ourselves beating our numerator on the frequency of a particular outcome and the denominator on the total frequency from the table.

The probability of a set of outcomes can be found from a frequency table by first summing the frequencies of those outcomes before calculating the probability.

And a table can also be used to classify outcomes into sets of outcomes or particular events as well.

Fantastic work today.

Thank you very much.