# Hannah’s Sweets

This was a question on this year’s Edexcel GCSE Mathematics examination. (Edited: Twitter)

Social media is often unpredictable.

Students taking the Edexcel GCSE Mathematics exam caused a consternation, divulging the difficulty of its questions.

One of these questions has been labelled *Hannah’s Sweets*, and reads as follows:

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.

Hannah then take at random another sweet from the bag.

She eats the sweet.

The probability that Hannah eats two orange sweets is [one third].

(a) Show that n² – n – 90 = 0.

It is a peculiar facet of mathematics that the amalgamation of two simpler concepts can seem difficult.

Exams are also flustering experiences.

## Conditional Probability

The initial equation can be arrived at through either conditional probability or combinatorics.

If Hannah takes a sweet from the bag, then 6 out of those n sweets are orange.

Now, the bag has 5 orange sweets and n – 1 sweets in total.

Thus, the probability of Hannah’s second choice also being an orange sweet, given her first was also orange, is 5 divided by n – 1.

Symbolically, this may be written:

This is elementary probability.

In the question, this probability is equal to one third, thereby giving the student an algebraic equation to rearrange.

A small amount of simple manipulation yields the answer.

## “Manipulate algebraic expressions”

Combinatorics may also reveal that algebraic equation, and avoid conditional probability.

Hannah must choose two orange sweets from a total of n sweets.

This is all the ways Hannah can choose two sweets out of the six orange sweets, divided by the total ways Hannah can choose two sweets from all n sweets.

Using binomial coefficients, this is typically said ‘6 choose 2’ divided by ‘n choose 2’.

GCSE Mathematics students should be competent and confident in both conditional probability and algebraic equations.

From the Edexcel GCSE Mathematics syllabus (2012), students should learn to:

List all outcomes for single events, and for two successive events in a systematic way and derive relative probability.

Furthermore, students are required to know how to:

Manipulate algebraic expressions by collecting like terms, by multiplying a single term over a bracket, and by taking out common factors.

A question of this kind is within the acceptable range of problems for students of this exam, whilst the fusion of these concepts can construct a confusing facade.

Exams are supposed to test.

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