# Always know how many sweets you have.

Time for a little silliness!

The Belfast Telegraph today is carrying a report on the bafflement caused by a question on EdExcel’s GCSE Maths paper and the viral reactions on Twitter.

The question was as follows:-

Hannah has 6 orange sweets and some yellow sweets.
Overall, she has n sweets.
The probability of her taking 2 orange sweets is 1/3.
Prove that: n^2-n-90=0

It’s not actually as difficult as it sounds.

The probability of drawing an orange sweet is the number of orange sweets over the total number of sweets, in other words 6/n.

Once Hannah has eaten the first orange sweet, there is one fewer sweet, and so the probability of drawing a orange sweet is 5/n-1.

The rules of probability are that if you want the chances of two things happening, you multiply their probabilities – so the chances of the first two sweets out of the bag being orange are (6/n)(5/(n-1)) and they equal 1/3.  (I’ll talk about some other probabilities in a minute.)

So take (6/n)(5/(n-1)) = 1/3 and multiply out the left hand side to get:
30/(n^2-n) = 1/3

If you multiply both sides in turn by the denominator of both sides (cross-multiplying), you get:
90 = n^2-n, or:
n^2-n-90=0

QED, and rather nerdily.  Worse, I ran it through the quadratic formula and I can tell you that unless Hannah lived in a parallel universe where you can have negative numbers of sweets, she had four yellow sweets, and n = 10.

I said I’d come back to other probabilities.  The first of these is the probability that at least one of the first two sweets would be orange.  You get these by adding the probabilities, but you have to add the right probabilities:

• The probability that the first sweet would be orange (6/10) and the second would be yellow (4/9) = 24/90, or 4/15
• The probability that the first sweet would be yellow (4/10) and the second would be orange (6/9) = 24/90, or 4/15
• The probability that both sweets would be orange, which we already know to be 1/3, or 5/15

That makes 13/15, which is pretty high.  To check, the final probability is that of drawing two yellow sweets (4/10 for the first and 3/9 for the second) is 12/90, or 2/15.

To complete the picture, in the parallel universe of negative numbers of sweets, Hannah would have had -15 yellow sweets for n=-9.  If you took those four sweets and an IOU for 15 more, shame on you. 😉