Year: 2021
Paper: 2
Question Number: 11
Course: LFM Stats And Pure
Section: Conditional Probability
No solution available for this problem.
Candidates were generally well prepared for many of the questions on this paper, with the questions requiring more standard operations seeing the greatest levels of success. Candidates need to ensure that solutions to the questions are supported by sufficient evidence of the mathematical steps, for example when proving a given result or deducing the properties of graphs that are to be sketched. In a significant number of steps there were marks lost through simple errors such as mistakes in arithmetic or confusion of sine and cosine functions, so it is important for candidates to maintain accuracy in their solutions to these questions.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
A train has $n$ seats, where $n \geqslant 2$. For a particular journey, all $n$ seats have been sold, and each of the $n$ passengers has been allocated a seat.
The passengers arrive one at a time and are labelled $T_1, \ldots, T_n$ according to the order in which they arrive: $T_1$ arrives first and $T_n$ arrives last. The seat allocated to $T_r$ ($r = 1, \ldots, n$) is labelled $S_r$.
Passenger $T_1$ ignores their allocation and decides to choose a seat at random (each of the $n$ seats being equally likely). However, for each $r \geqslant 2$, passenger $T_r$ sits in $S_r$ if it is available or, if $S_r$ is not available, chooses from the available seats at random.
\begin{questionparts}
\item Let $P_n$ be the probability that, in a train with $n$ seats, $T_n$ sits in $S_n$. Write down the value of $P_2$ and find the value of $P_3$.
\item Explain why, for $k = 2, 3, \ldots, n-1$,
\[
\mathrm{P}\bigl(T_n \text{ sits in } S_n \mid T_1 \text{ sits in } S_k\bigr) = P_{n-k+1},
\]
and deduce that, for $n \geqslant 3$,
\[
P_n = \frac{1}{n}\Biggl(1 + \sum_{r=2}^{n-1} P_r\Biggr).
\]
\item Give the value of $P_n$ in its simplest form and prove your result by induction.
\item Let $Q_n$ be the probability that, in a train with $n$ seats, $T_{n-1}$ sits in $S_{n-1}$. Determine $Q_n$ for $n \geqslant 2$.
\end{questionparts}
Some candidates misread the first part of the question and therefore attempted to solve a different question than was intended. The most common such misunderstanding that the probability P_n introduced in part (i) related specifically to a train with n seats. Where candidates did not have this problem the computations were done well. The explanation in part (ii) was also done well by most candidates who engaged meaningfully with the question. The deduction in this part of the question caused some trouble, but many were able to successfully complete this part. In particular, the reindexing of the sum within this solution was often overlooked or poorly explained. In part (iii) many candidates were able to identify the correct simplified form. However, there was some confusion about the difference between weak and strong induction meaning that many candidates were not able to give a satisfactory explanation of how the conclusion is drawn for the final mark in this section. Of those who successfully completed (iii) many were able to make good progress on the final part of the question.