Video: AQA GCSE Mathematics Higher Tier Pack 2 • Paper 2 • Question 6

𝜉 = {4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48}. 𝐴= square numbers. 𝐵 = multiples of 8. (a) Complete the Venn diagram. (b) Simon chooses one of the numbers at random. What is 𝑃(𝐴 ∪ 𝐵)?

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Video Transcript

We have a set of numbers: four, eight, 12, 16, 20, 24, 28, 32, 36, 40, 44, and 48. 𝐴 is equal to square numbers. 𝐵 is equal to multiples of eight. Part a) Complete the Venn diagram. Part b) Simon chooses one of the numbers at random. What is the probability that it comes from 𝐴 union 𝐵?

So the first part we need to do here is complete the Venn diagram. So to be able to complete the Venn diagram, we need to think which ones of our numbers are first 𝐴, square numbers, then 𝐵, multiples of eight. So let’s start with the square numbers. So to remind us what square numbers are, a square number is a result of a number being multiplied by itself. So for example, one multiplied by one equals one, so one is a square number. Two multiplied by two is equal to four, so four is a square number.

So therefore, if we take a look at our set of numbers, our first square number is going to be four. The next square number would be three multiplied by three, which would be nine. So nine is in there. But four multiplied by four is 16, and 16 is there. Then the next square number will be five multiplied by five, which would be 25. Well that’s not there. And then we’ve got six multiplied by six, which is 36. Well that’s one of our numbers. And then seven multiplied by seven is 49, but we haven’t got that. And that’s past the last number in our set. So therefore, we can say that set 𝐴 is four, 16, and 36.

So now what we need to do is find set 𝐵, which is the multiples of eight. Well the multiples of eight are the numbers that are in the eight times table. So the first one will be eight, and that’s in our set of numbers. Well the next one will be 16 because eight multiplied by two is 16; that’s in our set of numbers. It’s worth noting at this point it doesn’t matter if it’s in both sets; that’s allowed. And we will deal with that when we put them into the Venn diagram. Then the next multiple of eight is 24 cause three multiplied by eight is 24. And then we have 32, which again is in our set of values. And then we have 40 because eight times five is 40. And then finally we have 48. So we’ve now got our set 𝐵. So we’ve got our multiples of eight. And that’s eight, 16, 24, 32, 40, and 48.

So now what we want to do is put both sets into our Venn diagram. So we’re gonna start with 16. And we start with 16 because 16 is in fact in both of our sets. So this goes into the Venn diagram in the overlap section, which is called the intersection. And being in this region tells us that it’s in both set 𝐴 and set 𝐵. And now what we need to do is put the other values into our set 𝐴 circle and our set 𝐵 circle. So for set 𝐴, we’ve put in four and 36 into the left-hand circle, remembering not to be into the intersection area because they’re not also in set 𝐵. And then we’ve put the other values from set 𝐵 into the 𝐵 circle on the right-hand side of the Venn diagram.

So now if we look back at our original data set, we can see that we have some values that neither in set 𝐴, cause they’re not square numbers, nor set 𝐵 because they’re not multiples of eight. And they are 12, 20, 28, and 44. And these values go round the outside of our Venn diagram because what it tells us is that they are in our set of numbers; however, they’re not in set 𝐴 nor set 𝐵. So now we’ve completed part a as we’ve drawn the Venn diagram, okay? So now let’s move on to part b.

So to solve part b, we need to know what the bit of notation means. We’ve got 𝑃 and then we’ve got 𝐴 and then it looks like 𝐴 u 𝐵. And what this means is the probability of 𝐴 union 𝐵. And union means or when we’re look at the probability. So it says what’s the probability that is 𝐴 or 𝐵. And when we’re looking at a Venn diagram, if we’re thinking about the union between the two different sets, then it’s actually this shaded in area that I’ve done in that sketch here. So in our Venn diagram, it’s the probability that it’s any of these values I’ve circled in orange. So therefore, we consider that the probability of 𝐴 union 𝐵 is gonna be equal to eight over, and we’ve got eight over because there are eight values in set 𝐴 and 𝐵, and then it’s 12 as the denominator because there were 12 values in total.

So therefore, we know the probability of 𝐴 union 𝐵 is eight-twelfths. And we can simplify that by dividing numerator and denominator by four. So we can say that the probability of 𝐴 union 𝐵 is equal to two-thirds. And that’s because we have two-thirds of the values are in either set 𝐴 or set 𝐵.

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