2023 DHS P2 Q8

TGM adapted model: one round consists of exactly eight moves from \(X\) on a square lattice. The diagram shows the part of the lattice containing \(X\), \(Y\) and \(A\) to \(E\).

• The counter can only either move up or to the right along the grid lines.

• At every movement, including the first from \(X\), the player randomly moves the counter to the right with probability \(p\) or up with probability \(q\), \(p+q=1\). The moves taken at each junction are independent.

• A round is won if the counter passes through \(Y\) and is at \(B\) after its eighth move. Reaching the right-hand boundary earlier does not end the round; moves continue on the lattice. Every other final position is a loss.

1. Show that the probability of winning a round is \(30{{p}^{5}}{{q}^{3}}\).[3]

It is given that \(p=\frac{4}{5}\). Wang plays one game which comprises \(15\) rounds. Each round is independent of one another. Let the random variable \(W\) be the number of rounds that Wang wins.

2. Find the probability that Wang wins at least one-third of the \(15\) rounds.[1]
3. Wang plays \(40\) such games independently. Estimate the probability that the mean number of rounds won per game is at most one. Use the central limit theorem without a continuity correction.[3]

TGM adaptation note: the original source does not specify what happens on first reaching the right-hand boundary. The eight-move rule above makes the retained \(30p^5q^3\) calculation valid. The final central-limit-theorem task is optional extension beyond IB AA SL and HL.