How do I solve Hannah’s sweet question?

This refers to a question in the Edexcel GCSE paper this year which took students to Twitter venting their frustration. The question is as follows:

‘There are n sweets in a bag. 6 of the sweets are orange. The rest of the sweets are yellow.

Hannah takes a random sweet from the bag. She eats the sweet.

Hannah then takes at random another 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’

Seemingly out of nowhere you’re asked to prove that a certain quadratic equation holds using the information provided. The first three lines set up the situation whilst the fourth line provides you with some extra information to use to obtain the answer. Intuition should tell you that you need to calculate the probability that Hannah eats two orange sweet using the first three lines and then apply what you’re given in the fourth line.

So let’s do that. What’s the probability that the first sweet she eats from the bag is orange? There are n sweets in the bag, 6 of which are orange. So the probability is 6/n.

What’s the probability that the second sweet she eats from the bag is orange? Now there are n-1 sweets in the bag, 5 of which are orange (since she has eaten an orange sweet!). So the probability is 5/(n-1).

These two events are separate from one another, so the probability that both happen (i.e. both the sweets are orange) are the two probabilities multiplied together: 6/n × 5/(n-1) = 30/n(n-1)

But you’re told that this probability is 1/3! So all you need to do is set the expression equal to 1/3, rearrange and (hopefully!) obtain the required quadratic equation.

30/n(n-1) = 1/3

⇒ 90/n(n-1) = 1           (multiplying both sides by 3)

⇒ 90 = n(n-1)               (multiplying both sides by n(n-1))

⇒ n(n-1) – 90 = 0         (subtracting 90 from both sides)

⇒ n² – n – 90 = 0         (expanding the brackets)

Tah-dah. We’ve found the required equation and we’re done. This question was only worth three marks; a bit stingy in my opinion!

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