Hannah's Sweets

Quite a few GCSE students were stumped by a question about Hannah's sweets on yesterday's higher tier Edexcel paper.

The question was:

Hannah has 6 orange sweets and some yellow sweets.

Overall, she has n sweets. She takes one sweet from the bag and then another.

The probability of her taking 2 orange sweets is 1/3.

Prove that: 

The question is not actually that difficult if you remember how tree diagrams work.

When Hannah first takes a sweet there are 6 orange sweets out of n, so the probability of her choosing orange is 6/n.

When she chooses her second sweet there are now only 5 orange (if she chose orange the first time) out of a total of n-1 sweets.

You multiply the probabilities along the branches of a tree diagram, so

 Multiplying the 2 fractions on the left gives:

Cross-multiplying gives:

A-level maths now most popular subject

Pupils in England, Wales and Northern Ireland receive their A-level results today and they appear to have fallen slightly this year.

The Joint Council for Qualifications, issuing the results, said there has been a slight fall in A* and A grades and the overall pass rate is down for the first time in over 30 years. The percentage gaining the very highest A* grade has risen from 7.6% to 8.2%. 8.5% of boys' grades were A*, with girls' grades at 7.9%.

A-Level results this year are "broadly stable".

For the third successive year overall A* and A grades have fallen slightly (this year down from 26.3% to 26%), but exam officials are saying A-level results are broadly "stable".

For school leavers planning to go to university, there are suggestions this could be an unusually good year to apply. There are a record number of university places on offer this year - over 500,000 for the first time, which is a rise of over 30,000. Students may still get places even if they have missed their grades. The Ucas admissions service says initial figures show a 2% increase in students getting their first choice place.

Education Secretary Nicky Morgan says the government is "lifting the cap on aspiration". Universities Minister Greg Clark says the increase in the number of places is an "important source of social mobility".

There is a trend for more students to take so-called "facilitating subjects" at A-level, such as maths and physics, which can help university applications. Maths is now the most popular subject, overtaking English this year for the first time.

It is the first set of results following the Government's scrapping of January A-Level sittings. However, the fewer opportunities to take modules does not seem to have affected students' overall performance too badly.

Regarding other proposed changes, Labour's shadow education secretary Tristram Hunt said he would reverse the government's plan to remove the link between AS and A-levels. This de-coupling of the two exams would limit young people's "opportunity to realise their full potential", said Mr Hunt.

Would you like to share your results story with us? Comment on this article, or email info@mathsbank.co.uk.

GCSE Results Fall for the First Time

Over a million students in England, Wales and Northern Ireland receive their GCSE results today and for the first time ever, overall results have fallen.

Since replacing the old O-Level and CSE exams in 1988, the percentage of GCSE papers awarded an A*-C grade has been rising inexorably, year on year.

Today's GCSE results show that 69.4% of entries earned grades A*-C, compared with 69.8% last year.

The proportion of pupils receiving the top grades (A and A*) has also fallen for the first time, down from 23.2% to 22.4%.

As with the A-Level exams, England's exams regulator, Ofqual, has told exam boards they would have to justify the overall pass rate if it were significantly different from previous years. This has happened in an attempt to address concerns of "dumbing down" and to ensure results were comparable,

However, this system, known as "comparable outcomes", has created a furore in itself. There have been accusations that the exam boards had been asked by the Department for Education to fix the results.

Teaching unions are not particularly happy about any drop in apparent standards. They claim that any rise in grades is the result of better teaching and better-prepared students.

In most subjects, including maths, there was a fall in the overall proportion of A*-C grades, but the fall was particularly marked in English. Many teachers claim that students have been marked down by an entire grade in English compared with the results that teachers had predicted.

Countering this, the Joint Council for Qualifications, which publishes the annual results, said that the fall in A*-C English results was partly down to more candidates sitting the exam during the winter exam season.

This set of results is set against the backdrop of considerable changes being planned for GCSEs. As of September, the GCSE exams set by the English exam boards will no longer be modular. Instead, students will sit a single final exam. In addition, GCSE exams in English literature, geography, history and religious education will be assessed for spelling, punctuation and grammar. These new strictures will affect those sitting their GCSEs in the summer of 2014.

Schools in Wales and Northern Ireland will continue to be able to opt for modular GCSEs.

Eventually, Michael Gove has expressed an interest in returning to an exam more similar to the old O-Level. Some in education are even questioning whether it is necessary to take formal exams at all at the age of 16.