Year: 2020
Paper: 2
Question Number: 11
Course: LFM Stats And Pure
Section: Conditional Probability
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
A coin is tossed repeatedly. The probability that a head appears is $p$ and the probability that a tail appears is $q = 1 - p$.
\begin{questionparts}
\item A and B play a game. The game ends if two successive heads appear, in which case A wins, or if two successive tails appear, in which case B wins.
Show that the probability that the game never ends is $0$.
Given that the first toss is a head, show that the probability that A wins is $\dfrac{p}{1 - pq}$.
Find and simplify an expression for the probability that A wins.
\item A and B play another game. The game ends if three successive heads appear, in which case A wins, or if three successive tails appear, in which case B wins.
Show that
\[\mathrm{P}(\text{A wins} \mid \text{the first toss is a head}) = p^2 + (q + pq)\,\mathrm{P}(\text{A wins} \mid \text{the first toss is a tail})\]
and give a similar result for $\mathrm{P}(\text{A wins} \mid \text{the first toss is a tail})$.
Show that
\[\mathrm{P}(\text{A wins}) = \frac{p^2(1-q^3)}{1-(1-p^2)(1-q^2)}.\]
\item A and B play a third game. The game ends if $a$ successive heads appear, in which case A wins, or if $b$ successive tails appear, in which case B wins, where $a$ and $b$ are integers greater than $1$.
Find the probability that A wins this game.
Verify that your result agrees with part \textbf{(i)} when $a = b = 2$.
\end{questionparts}
This was the more popular of the probability questions and many good attempts were seen, although the majority were incomplete and only attempted the first one or two parts. Some candidates made errors when dealing with the conditional probabilities, often thinking for example that P(A wins) could be obtained by adding P(A wins|H first) and P(A wins|T first). In general, those that were able to work confidently with the conditional probabilities were able to perform very well on this question. In the first part, a number of candidates failed to consider that games could begin with either heads or tails when showing that the probability that the game never ends is 0. Additionally, some candidates assumed that p = q = 1/2 for the first part of the question, although they often did then use correct expressions in terms of p and q in the later parts of the question. In part (ii) some candidates tried to find a way to enumerate all possible sequences for any total number of flips, but this approach almost always resulted in some of the possible cases being omitted. Many candidates failed to spot that the solution to the third part could be found by an analogous method to that used in part (ii) and so in many cases no attempt was made at this final part.