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 Results Show Slight Fall in Top Grades

More than 300,000 A-Level students in England, Wales and Northern Ireland have found out their A-level and AS results today, and join the scramble for university places.

The results, published by the Joint Council for Qualifications (JCQ), show that there has been a fall in the proportion of A-levels awarded top grades for the second year in a row, after three decades of steady increases. 26.3% of all entries were given A or A* grades this year, a slight fall from 26.6% in 2012. Previously, the proportion getting top grades had risen each year since 1980.

The national pass rate rose marginally to 98.1%. This has also risen for about 30 years.

Continuing recent trends, more students are taking A-levels in maths and science and there is a continued fall in those taking languages. Maths rose by just under 3% and further maths by 4.5%.

Girls are still ahead of boys when looking at the top grades, A or A* (26.7% for girls compared with 25.9% for boys), but boys this year were more likely to get the A* grade (7.9% of boys' entries, compared with 7.4% for girls).

The university admissions body Ucas has said that 385,910 students have already been accepted by UK universities, 31,600 more than at the same point last year.

The UK government claims its reforms to make it easier for universities to take on the students that they want to recruit have sped up the process of accepting students. Under these changes, universities in England are being allowed to admit as many top-performing students (gaining ABB or more) as they want to. For students with lower results, universities are allocated a quota of undergraduates they can recruit. Last year, thousands of course places were left unfilled.

The change was introduced to allow the most popular universities to expand. It came in alongside higher tuition fees, which rose to a maximum of £9,000 a year from autumn 2012.

Students in Northern Ireland continue to perform best. 83.5% of entries here scored between an A* and a C and 30.7% were awarded the top grades of A or A*. In Wales, these figures are 75.2% and 22.9% respectively and in England 77% and 26.3%.

From 2015 the government plans to introduced major changes to A-levels. The AS-level will no longer count towards the final A-level grade and, with modules being phased out, all exams will be taken at the end of the two year course.

Congratulations to all those who gained the grades they were looking for today - and best of luck to everybody looking for a university place.

A-Level Results Day. Top Grades Down.

It's that time of year again. A-Level results came out this morning, with many students in England, Wales and Northern Ireland able to find out their grades online from 7 a.m.

The headline from today's set of results is that the percentage of students being awarded the top grades (A and A*) is down, for the first time in 20 years. Today, 26.6% of A-level entries achieved the top two grades, compared with 27% last year. The overall pass rate, meanwhile, continues to rise, for a 30th consecutive year.

The drop in the percentage of top grades was particularly pronounced in Northern Ireland, although overall the performance there is still better than average, with 31.9% of entrants being awarded top grades, down from last year's figure of 34.5%, according to figures from the Northern Irish exam board CCEA.

Since 2010, the exam watchdog Ofqual has been telling exam boards they must be able to account for any upward movement in grades, to show that it was a result of a real improvement in performance.

Girls continue to gain slightly better grades than boys, but this year boys obtained more A* grades, with 8% of entries gaining the top grade, compared with 7.9% for girls.

Once again, maths and science subjects proved popular, with increases in the number of entrants, while the trend for modern languages is still downwards.

For many of the 330,000 students getting their results today, A-Level results will determine whether they can go to their chosen university.

The number of applications for university places is about 7% down this year, following the rise in tuition fees to a maximum of £9000 in England. Students in other parts of the UK may not have to pay so much. Northern Irish students staying in Northern Ireland will face no rise in fees. The Welsh government has agreed to subsidise students facing higher fees in other parts of the UK. Scottish students, who do not sit A-Levels, pay no fees at all if they choose to attend Scottish universities.

Students are Revolting. Again.

Things are changing fast.

Sixth formers demonstrate at rises in university tuition fees
Photo from the Guardian

Students are on the move again. All across the country during November, there have been at least two days of protests, marches and sit-ins, sometimes peacefully, sometimes a little more forcefully. More protests are planned for the coming weeks.

We all know the cause of the unrest. The Government plans to allow universities to increase tuition fees. There is currently a cap of £3290 per year, but this will be removed from September 2012, replaced with a maximum of £9000 per year. Any universities wishing to charge more than £6000 per year will have to put in place measures, such as offering bursaries, to encourage students from poorer backgrounds to apply. All of these changes come about because of recommendations in the Brown Review, published in October.

You can see the Guardian's photo coverage of the demonstrations here:

http://www.guardian.co.uk/education/gallery/2010/nov/24/tuition-fees-students

and here:

http://www.guardian.co.uk/education/gallery/2010/nov/24/tuition-fees-universityfunding

On this blog we've reported on the government's plans for reform several times:

http://blog.mathsbank.co.uk/2010/08/another-government-another-shake-up.html

http://blog.mathsbank.co.uk/2010/10/help-i-cant-afford-to-go-to-uni.html

http://blog.mathsbank.co.uk/2010/09/free-schools-have-fifteen-minutes-of.html

For those of us of a certain age, all of this will bring back memories of a previous set of demonstrations. In the late 1980s and early 1990s students across the country brought universities to a standstill over plans to introduce the first student loans. The size of these loans was minuscule compared with those students take out today. But the demonstrators would probably say they were vindicated, in that those loans set a precedent for and led directly to the massive debts inflicted on the student population today.

One striking difference between these protests and those of 20 years ago, however, is the involvement of school pupils. A-Level students are fully aware of the implications of the cuts on their futures, not just over the next three years, but, if the plans go ahead, for years and decades to come. The debt levels some will accrue to gain a university education are truly mind-boggling.

David Cameron is of course a prime target for blame. But Nick Clegg is taking a huge amount of criticism as well. Before the general election in May, the Liberal Democrats pledged to oppose any move to increase tuition fees. Every member of the party signed a pledge to honour this commitment and vote against any such proposal. So understandably, ever since leading his party into a coalition with the Conservatives, Clegg has been attacked for abandoning these principles. Recently, however, it emerged that the Lib Dems had considered abandoning the commitment even before the election. Talking to students and teachers, it is this that has enraged them more than anything else.

Now there is news that the government wants to cut funding to school sixth forms as well.

http://www.bbc.co.uk/news/education-11845857

Head teachers are warning of the drastic effects such changes could bring about, including loss of facilities and equipment, even teaching staff and an overall deterioration in the quality of A-Level teaching. Once again, on this blog, we put the case that, although we undoubtedly live in difficult times, these measures, both the university tuition fee rises and the cuts in sixth form budgets, are going far too far, far too fast.