Lesson video

Hello, I'm Mr. Langton.

And today we're going to look at more problems involving theoretical probability and relative frequency.

All you going to need is something to write with and something to write on.

Try and find a quiet space where you won't be disturbed.

And when you're ready, we'll begin.

We'll start off with the try this activity.

How many times do you expect to get the purple on the spinner? If I did two spins or five spins, eight spins, 12 spins or 80 spins.

Pause the video, and work out your answers.

When you're ready unpause it and we can go through it together.

You can pause in three, two, one.

Okay.

How did you get on? I'd start off by looking at the spinner itself.

There are three purple parts and two green parts.

So the probability of getting a purple is 3/5.

If I spun that spinner twice and 3/5 of the time, I'm expecting it to come up purple.

That's going to come out to 6/5.

So I would say approximately once I'm going to expect it to come purple.

Let's make five times.

That's going to get me, sorry, 15/5 which is three times.

If I spin it eight times 3/5 multiply by 8 that's going to give me 24/5 which is 4.

8, so that's approximately equal to five.

If I was to spin it 12 times 3/5 of 12, there's going to be 36 over five which is a little bit bigger than seven.

So it's about seven times.

And if I spent it 80 times, now I'm doing 3/5 of 80, which is 48 exactly.

So hopefully, you got answers similar to mine.

So one time, in fact, I'm not going to say equals, I'm going to say approximately one time.

Approximately three times.

Approximately five times.

Approximately seven times and approximately 48 times.

Cala buys has a new book every week.

Her favourite types are Biography, Romance, Horror, and Poetry.

The table shows the probability that Cala will select a particular type of book.

Part a.

What is the probability that Cala will choose a Horror book?.

Now we're not given the probability for a horror book but we're told that she only buys one of these four types of books.

Now, if we add together, the three probabilities that we know, that's 0.

28, 0.

12 and 0.

2 my letter is a little bit small but you got to be really careful to line them up properly.

So where the decimal point is on you tenths and your hundredths So, eight and two is 10.

Two, one, three, four, five, six.

So the total probability of the three that we know is 0.

6.

And because we know that all the possible outcomes shoot up to a whole one.

If we do a whole one, take away 0.

6, that will tell us the probability that she'll take a horrible, 0.

4.

All right.

Write that as well.

Next to the answer.

B.

How many Romance books do you think she will buy over 10 weeks? Well, the probability that she buys a romance book, is 0.

2.

If you're going to go 10 times.

10 weeks are expected to be buying two Romance books over that time.

Now, for the horror books she's buying over 25 weeks? The probability of getting a horror book, is there a 0.

4? She can buying books 25 times.

And so I would expect that over the 25 weeks, 10 of those times she's going to buy a horror book.

Okay.

Now it's your turn.

Pause the videos, to adjust the worksheet when you're done unpause it.

We can go through it together.

Good luck.

How did you get on? Let's have a look at the answers.

So for the first one, Benh has a bias dice.

The probability of it landing on a six is 2/3.

She rolls the dies 300 times.

How many times do you think she'll roll a six? So she does 300 times and two thirds of the time it should land on a six.

So we're going to estimate there's going to be 200 times.

Underneath that one.

Yasmin has a bag of mixed bulbs, which he intends to plant in her garden.

The packaging states that the probability that each bulb is a certain type of species of flower.

Comes up below that.

Yasmin plants 250 bulbs.

How many of each species would you expect to grow? So in each case, we're going to multiply the probability by 250.

O.

4 times 250.

0.

12 times 250 and 0.

32 multiplied by 250.

Now I don't have a calculator to hunt.

So I'm going to show you a little trick to multiply by 250.

250 is a quarter of 1000.

So if I multiply number by a thousand and then find a quarter of that, that'll be the same as multiplying by 250 or the other way around.

I could find a quarter of it first and then multiply by 1000.

In the case of the daffodils, I'm just going to change colour.

In the case of the daffodils, 0.

16, a quarter of that is 0.

04.

I need to multiply that by 1000 which is going to be 0.

4.

40.

Right.

So rose, 0.

4.

So if I divide that by four, if I find a quarter and get the 0.

1, which I need to times by 1000.

Times is by 10 is one, times by 100 will be 10.

So times that by another 10.

And that's going to be 100 roses.

That looks pretty good so far for the two, because we know the total number is 250.

We know there would be more roses than daffodils.

So our method seems to be working so far, right? Alliums 0.

12, a quarter of that 0.

03 times 1000.

And so that's going to be 30.

Iris 0.

32, so divide that by four.

And I get 0.

08, which I'm going to multiply by 1000 to get 80.

Now let's just double check that these numbers add up to make 250.

So 40 and 100 is 140, add 30 is 170, add 80 to make 250.

So yup.

I'm really happy that I've got that one right.

Onto the right hand side.

Xavier has a biassed coin.

The probability that it will land on a head is 0.

3.

He plans to flip it to 150 times.

How many times you think it will land on a tail? The first thing to notice is that we're told the probability that it will land on a head.

Only we've got to calculate the probably it will land on a tail.

So if it's 0.

3 transit to landing on a head, that's going to be a 0.

7 chance that it lands on a tail.

He's doing the experiment 150 times.

So we need to do 0.

7 multiplied by 150.

That's not the sort of thing that I can do in my head but I can make it, well.

Do you know what.

I can make it a little bit easier just by changing around what I'm doing.

0.

7 times 150.

If I made the 0.

7, 10 times bigger, and I make the 150, 10 times smaller.

So those answers will be equivalent.

And now seven five, 35, seven ones is a seven.

So this is going to be 105 times.

Finally, Antoni does an experiment.

He throws a dice 1,200 times, He scores a four, 200 times, Do you think that the diocese fair? If it was a fair dice, the probability of landing on a four would be 1/6 all the time.

Now, if he's done 1,200 experiments and 200 times he got a four, let's simplify that fraction, we can divide both by 100 to get 2/12.

And two twelfths is 1/6.

So yes, I think the dice is fair.

We'll finish with the Explorer activity.

Xavier and Yasmin have tried choosing cubes out of the bag and then replacing them.

They know that there are 10 cubes all together and that there are the green or white.

They run four experiments and the results are shown in the table.

Each of this statements I'd like to know if you agree with them or if you disagree with them.

You can give a reason as to why or why not.

Pause the video and have a go yourself.

When you're ready, unpause it, and we can discuss the answers together.

In pausing three, two, one.

Okay.

I'll give you my thoughts.

So Xavier says: That there I must be more green cubes since green was chosen more often in three of the four experiments.

That seems like a fair point to me it's pretty close.

But when they did five trials there were more greens than there were whites.

When they did 10 trials, there're more greens than there are whites.

And when they did 15, there're more greens that there are whites.

Only just but there was always more.

It was only that last one, when they did 20 trials, actually it only came seven times compared to 13 for whites.

Now, this is why Yasmin's statement has much more power for me: Experiment four had the most trials, so this was the best.

There must be more white cubes.

It's still fairly close.

It would have been 50/50 on the last one, had just three times she chose a green instead of white.

So that would have been, it's close, but the more trials that you do, the more accurate you're likely to be for your final answer.

And so I'm more likely, I'm more inclined, to agree with Yasmin than I'm with Xavier.

Xavier could be right, especially with those 15 trials, eight compared to seven.

I think the numbers are pretty close in each one.

But the more trials you do the better and that's the one that I'd be leaning towards.

So I think that Yasmin is more likely to be correct.

We'll finish that for today, now.

I'll see you later.

Goodbye.