Year: 2009
Paper: 1
Question Number: 12
Course: LFM Stats And Pure
Section: Tree Diagrams
No solution available for this problem.
There were significantly more candidates attempting this paper again this year (over 900 in total), and the scores were pleasing: fewer than 5% of candidates failed to get at least 20 marks, and the median mark was 48. The pure questions were the most popular as usual; about two-thirds of candidates attempted each of the pure questions, with the exceptions of question 2 (attempted by about 90%) and question 5 (attempted by about one third). The mechanics questions were only marginally more popular than the probability and statistics questions this year; about one quarter of the candidates attempted each of the mechanics questions, while the statistics questions were attempted by about one fifth of the candidates. A significant number of candidates ignored the advice on the front cover and attempted more than six questions. In general, those candidates who submitted answers to eight or more questions did fairly poorly; very few people who tackled nine or more questions gained more than 60 marks overall (as only the best six questions are taken for the final mark). This suggests that a skill lacking in many students attempting STEP is the ability to pick questions effectively. This is not required for A-levels, so must become an important part of STEP preparation. Another "rubric"-type error was failing to follow the instructions in the question. In particular, when a question says "Hence", the candidate must make (significant) use of the preceding result(s) in their answer if they wish to gain any credit. In some questions (such as question 2), many candidates gained no marks for the final part (which was worth 10 marks) as they simply quoted an answer without using any of their earlier work. There were a number of common errors which appeared across the whole paper. These included a noticeable weakness in algebraic manipulations, sometimes indicating a serious lack of understanding of the mathematics involved. As examples, one candidate tried to use the misremembered identity cos β = sin √(1 − β²), while numerous candidates made deductions of the form "if a² + b² = c², then a + b = c" at some point in their work. Fraction manipulations are also notorious in the school classroom; the effects of this weakness were felt here, too. Another common problem was a lack of direction; writing a whole page of algebraic manipulations with no sense of purpose was unlikely to either reach the requested answer or gain the candidate any marks. It is a good idea when faced with a STEP question to ask oneself, "What is the point of this (part of the) question?" or "Why has this (part of the) question been asked?" Thinking about this can be a helpful guide. One aspect of this is evidenced by pages of formulæ and equations with no explanation. It is very good practice to explain why you are doing the calculation you are, and to write sentences in English to achieve this. It also forces one to focus on the purpose of the calculations, and may help avoid some dead ends. Finally, there is a tendency among some candidates when short of time to write what they would do at this point, rather than using the limited time to actually try doing it. Such comments gain no credit; marks are only awarded for making progress in a question. STEP questions do require a greater facility with mathematics and algebraic manipulation than the A-level examinations, as well as a depth of understanding which goes beyond that expected in a typical sixth-form classroom.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1501.5
Banger Comparisons: 2
Prove that, for any real numbers $x$ and $y$, $x^2+y^2\ge2xy\,$.
\begin{questionparts}
\item Carol has two bags of sweets. The first bag contains $a$ red sweets
and $b$ blue sweets, whereas the second bag contains $b$ red sweets and
$a$ blue sweets, where $a$ and $b$ are positive integers. Carol shakes
the bags and picks
one sweet from each bag without looking. Prove that the probability
that
the sweets are of the same colour
cannot exceed the probability that
they are of different colours.
\item Simon has three bags of sweets. The first bag
contains
$a$ red sweets, $b$ white sweets and $c$ yellow sweets, where $a$, $b$ and
$c$ are positive integers. The second
bag contains
$b$ red sweets, $c$ white sweets and $a$ yellow sweets. The third
bag contains
$c$ red sweets, $a$ white sweets and $b$ yellow sweets.
Simon shakes the bags and
picks one sweet from each bag without looking.
Show that the probability that exactly two of the sweets are of the
same colour is
\[
\frac {3(a^2b+b^2c+c^2a+ab^2 + bc^2 +ca^2)}{(a+b+c)^3}\,,
\]
and find the probability that the sweets are all of the same colour.
Deduce that the probability that exactly two of
the sweets are of the same colour is at least 6 times the probability
that the sweets are all of the same colour.
\end{questionparts}
This was the more popular of the probability questions, and was attempted by almost a quarter of the candidates. It gained the highest scores on the whole paper: the median mark was 14. The introductory part caused some problems. This is a standard result which should have been recognised as such. Some candidates began with (x + y)² and became stuck. Others attempted to prove the result by induction, which will be challenging as x and y are any real numbers. Part (i) was generally done extremely well, with many candidates drawing an appropriate tree diagram. However, some candidates failed to add the probabilities (more commonly when they had not drawn a tree diagram), asserting things such as P(same colour) = ab/(a+b)² instead of 2ab/(a+b)². Part (ii) caused more problems. Many candidates decided that it would be useful to expand (a + b + c)³ and wasted a lot of time doing so. Another common issue was not adequately justifying the probability stated in the question; the arguments proposed for counting the number of possibilities were often weak or confused. Students would have done well to either draw a tree diagram (possibly abbreviated) or to explicitly list all of the possibilities. While taking time, it would have ensured that they reached the correct answer. Most of the candidates were able to calculate the probability of all three being the same colour correctly, although some left out the factor of 3. The last part of the question caused the most difficulty. Some suggested that since there were 18 ways of having exactly two the same colour and 3 ways of having them all the same colour, the former was 6 times more likely than the latter. This totally ignored the probabilities of these events and the work done earlier, as well as the "deduce" in the question. Others assumed a ⩽ b ⩽ c and then wrote a + x = b, a + y = c to try to relate it to the initial inequality, but struggled to get further. Nevertheless, there were a good number of students who correctly related the earlier inequality to the new context and succeeded in making the required deduction.