Year: 2020
Paper: 2
Question Number: 12
Course: LFM Stats And Pure
Section: Discrete Probability Distributions
No solution available for this problem.
There were just over 800 entries for this paper, and good solutions were seen to all of the questions. Candidates should be aware of the need to provide clear explanations of their reasoning throughout the paper (and particularly in questions where the result to be shown is given in the question). Short explanatory comments at key points in solutions can be very helpful in this regard, as can clearly drawn diagrams of the situation described in the question. The paper included a few questions where a statement of the form "A if and only if B" needed to be proven – candidates should be aware of the meaning of such statements and make sure that both directions of the implication are covered clearly. In general, candidates who performed better on the questions in this paper recognised the relationships between the different parts of each question and were able to adapt methods used in earlier parts when working on the later sections of the question.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
The score shown on a biased $n$-sided die is represented by the random variable $X$ which has distribution $\mathrm{P}(X = i) = \dfrac{1}{n} + \varepsilon_i$ for $i = 1, 2, \ldots, n$, where not all the $\varepsilon_i$ are equal to $0$.
\begin{questionparts}
\item Find the probability that, when the die is rolled twice, the same score is shown on both rolls. Hence determine whether it is more likely for a fair die or a biased die to show the same score on two successive rolls.
\item Use part \textbf{(i)} to prove that, for any set of $n$ positive numbers $x_i$ ($i = 1, 2, \ldots, n$),
\[\sum_{i=2}^{n}\sum_{j=1}^{i-1} x_i x_j \leqslant \frac{n-1}{2n}\left(\sum_{i=1}^{n} x_i\right)^2.\]
\item Determine, with justification, whether it is more likely for a fair die or a biased die to show the same score on three successive rolls.
\end{questionparts}
This question had the second smallest number of attempts on the paper. Many of these successfully completed the first part of the question, but then made little progress in the later sections. Part (i) was generally well done, although some candidates did not appreciate that ∑εᵢ = 0 when determining whether a fair or biased die is more likely to show the same score on two successive rolls. Where candidates were able to see the connection between the first and second parts of the question, the required result was generally proven clearly. Part (iii) could be approached in a similar way to part (i), but many of the candidates who reached this point failed to deal with the more complicated terms that arise from the expansion. Correct solutions to this part were generally very well set out.