Year: 2008
Paper: 1
Question Number: 13
Course: LFM Stats And Pure
Section: Probability Definitions
No solution available for this problem.
There were significantly more candidates attempting this paper this year (an increase of nearly 25%), but many found it to be very difficult and only achieved low scores. The mean score was significantly lower than last year, although a similar number of candidates achieved very high marks. This may be, in part, due to the phenomenon of students in the Lower Sixth (Year 12) being entered for the examination before attempting papers II and III in the Upper Sixth. This is a questionable practice, as while students have enough technical knowledge to answer the STEP I questions at this stage, they often still lack the mathematical maturity to be able to apply their knowledge to these challenging problems. Again, a key difficulty experienced by most candidates was a lack of the algebraic skill required by the questions. At this level, the fluent, confident and correct handling of mathematical symbols (and numbers) is necessary and is expected; many students were simply unable to progress on some questions because they did not know how to handle the algebra. There were of course some excellent scripts, full of logical clarity and perceptive insight. It was also pleasing that one of the applied questions, question 13, attracted a very large number of attempts. However, the examiners were again left with the overall feeling that some candidates had not prepared themselves well for the examination. The use of past papers and other available resources to ensure adequate preparation is strongly recommended. A student's first exposure to STEP questions can be a daunting, demanding experience; it is a shame if that takes place during a public examination on which so much rides.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1452.7
Banger Comparisons: 3
Three married couples sit down at a round table at which
there are six chairs. All of the possible seating arrangements
of the six people
are equally likely.
\begin{questionparts}
\item
Show that the probability that each husband sits next to his wife
is $\frac{2}{15}$.
\item
Find the probability that exactly two husbands sit next to their
wives.
\item Find the probability that no husband sits next to his wife.
\end{questionparts}
This combinatorics question was attempted by close to half of all candidates, a very encouraging statistic. About two-thirds of the attempts did not progress very far, gaining five marks or fewer, but of those who did get further, the marks were fairly evenly distributed. For part (i), most attempts reached the stated answer, although a significant number used very creative, if inaccurate or meaningless, ways of doing so. The majority of candidates used counting methods, and many of these were successful to a greater or lesser extent. The other method used by many candidates was to consider the probability of the first wife sitting next to her husband (2/5) and the conditional probability of the spouse of the other person sitting next to the first husband sitting next to them (this is 1/3), and then multiplying these. It is crucial at this point to reinforce that candidates must explain their reasoning in their answers, especially when they are working towards a given answer. Simply writing 2/5 × 1/3 = 2/15 is woefully inadequate to gain all of the available marks; there must be a justification of the reasoning behind it. Parts (ii) and (iii) were found to be a lot more challenging. A number of candidates attempted to construct probabilistic arguments, which are very challenging in this case. The successful attempts all used pure counting arguments. The examiners often found it challenging to decipher their thinking, though, as the explanations were often somewhat incoherent. Those who used counting arguments usually made good progress on both parts. The favoured method for part (iii) was to use P(no pairs) = 1 − P(≥ 1 pair). It would have certainly been worth checking the answer obtained using a direct method, as this would have caught a number of errors. The main errors encountered in good attempts at the later parts of the question were a failure to consider all possible cases or a miscounting of the number of ways each possible case could occur. Overall, this question was answered well by a significant number of candidates.