Remember the uproar on Twitter about an Edexcel GCSE Maths paper, sparking the #EdexcelMaths trend on Twitter? Whether you do or don't, the question that caused the controversy is within the GCSE syllabus and it is important to be able to tackle it. The question is as follows:
Q) There are n sweets in a bag. 6 of the sweets are orange. The rest of the sweets are yellow.
Hannah takes at random a sweet from the bag. She eats the sweet.
The probability that Hannah eats two orange sweets is 1/3
Show that n2 - n - 90 = 0
It is easy to be thrown off with this question because of the surprising ending. You may perhaps have been expecting a probability question and then you were asked a quadratic equation question. The truth is, if we work through this logically, we can use both our probability skills and our algebra skills to crack this.
First we need to find the probability that Hannah eats two orange sweets in terms of n.
The probability that Hannah eats an orange sweet first is the number of orange sweets in the bag (6, from the question) divided by the total number of sweets in the bag (n, again, from the question). So the probability of the first sweet being orange is 6/n.
Now we need to determine the probability of Hannah picking an orange sweet second from the bag, if the first she picked was also orange. To figure this out, we again do the number of orange sweets in the bag (5, as Hannah ate one), divided by the total number of sweets in the bag (n-1, again, as Hannah ate one). So the probability of Hannah picking a second orange sweet from the bag is 5/(n-1).
To figure out the probability of Hannah picking 2 orange sweets from the bag we multiply those 2 probabilities together.
6/n * 5/(n-1) = 30/n(n-1) = 30/(n2-n)
And we were told in the question that this probability was actually equal to 1/3. Therefore:
30/(n2-n) = 1/3
All we have to do now is manipulate this equation so that we get the one asked for in the question. First we can multiply both sides by 3:
90/(n2-n) = 1
Then we can multiply both sides by (n2-n):
90 = n2-n
And if we collect all the terms on one side we get:
Which is what we were asked to prove in the original question.
Hopefully you are able to follow this explanation, and if not, perhaps try running through it again step by step to try and understand it. There is no one skill used here that is particularly difficult - the challenge in this question is knowing how and when to apply the maths skills that you have, and this is a challenge that will keep facing you with maths if you intend to take it further.