In England, a number of students who took the GCSE mathematics test have been complaining about a question involving Hannah and her sweets. Here is the question:

There are sweets in a bag. Six 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 .

Not a difficult question. I would lengthen the last sentence to read: Use this information to show that must satisfy the following equation. But a pointless one. It gives the study of mathematics a bad name. How is it we know there are only two colors of sweets in the bag without knowing ? How is it we know that there are only 6 orange sweets without knowing how many yellow ones there are? Why can’t I work out by emptying the bag and counting its contents? In short, students are asked to accept an implausible premise to compute something that can be done simply by other means.

## 5 comments

June 5, 2015 at 12:02 pm

tjungbauWith all due respect: Europe is not the kingdom of professors putting much emphasis on the underlying story of a math question. In my experience such questions are significantly better in the US (or at my limited sample of top US universities). Might be caused by the fact that in the private money for education system complaints are both uttered more frequently and dealt with more seriously.

June 5, 2015 at 4:43 pm

rvohraThe exam is not, I think, set by Professors. Its for school students. This particular version of the GCSE exam, called the Edexcel, is administered by the Pearson company (a publishing and testing outfit).

June 5, 2015 at 10:43 pm

blinkWell said! The story here is often called “pseudo context” since it has no bearing at all on the problem — exactly the sort of device that gives mathematics a bad name and leaves students feeling it is a jumble of meaningless facts. Probably the test makers were able to use this question to check a box for “diversity” (for the female name) and “real-world setting” (even though it fails the sniff test).

June 11, 2015 at 11:09 am

Belinda WongThe probability of any particular sweet being orange is 6/n. That applies to each and every sweet whether it is in the bag or out of it. The probability of any two sweets both being orange is 36/(n x n). That probability can never equal 1/3, its impossible. The examiners made a mistake and its no wonder students were confused.

June 11, 2015 at 1:51 pm

rvohraNope, the examiners have not erred, at least in the computation. 36/n^2 is the probability of selecting two orange sweets in succession with replacement. The examiners are asking for the probability of selecting two orange sweets in succession without replacement.