Year: 2007
Paper: 1
Question Number: 12
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 50%), but many found it to be very difficult and only achieved low scores. In particular, the level of algebraic skill required by the questions was often lacking. The examiners' express their concern that this was the case despite a conscious effort to make the paper more accessible than last year's. At this level, the fluent, confident and correct handling of mathematical symbols (and numbers) is necessary and is expected; many good starts to questions soon became unstuck after a simple slip. Graph sketching was usually poor: if future candidates wanted to improve one particular skill, they would be well advised to develop this. There were of course some excellent scripts, full of logical clarity and perceptive insight. It was pleasing to note that the applied questions were more popular this year, and many candidates scored well on at least one of these. It was however surprising how rarely answers to questions such as 5, 9, 10, 11 and 12 began with a diagram. However, the examiners were left with the overall feeling that some candidates had not prepared themselves well for the examination. The use of past papers 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. Further, and fuller, discussion of the solutions to these questions can be found in the Hints and Answers document.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1484.0
Banger Comparisons: 1
\begin{questionparts}
\item A bag contains $N$ sweets (where $N \ge 2$),
of which $a$ are red. Two sweets are drawn
from the bag without replacement. Show that
the probability that the first sweet is red
is equal to the probability that the second sweet is red.
\item There are two bags, each containing $N$ sweets (where $N \ge 2$).
The first bag contains $a$ red sweets, and the
second bag contains $b$ red sweets. There is also a
biased coin, showing Heads with probability $p$ and Tails with probability $q$, where $p+q = 1$.
The coin is tossed. If it shows Heads then a
sweet is chosen from the first bag and transferred
to the second bag; if it shows Tails then a sweet
is chosen from the second bag and transferred
to the first bag. The coin is then tossed a second time:
if it shows Heads then a sweet is chosen from the first bag,
and if it shows Tails then a sweet is chosen from the second bag.
Show that the probability that the first sweet
is red is equal to the probability that the second sweet is red.
\end{questionparts}
Very few tree diagrams were seen here, and hence very few correct solutions were constructed; a clear tree diagram is invaluable when attempting a complicated probability question such as part (ii). Most candidates identified some (if not all) of the possible outcomes, but many mistakes were made (for example, writing a denominator of N rather than N + 1 or N – 1). The subsequent algebraic simplification was found to be very demanding. Candidates would have probably made more progress if they had been more willing to factorise groups of terms which had obvious common factors, rather than (for example) attempting to write all the fractions with a common denominator.