Navigation game through a checkpoint

One round of a navigation game is played by moving a token from the starting point \(P\) to one of the finishing points on the right-hand edge of the grid shown.

• The token may only move one grid spacing to the right or one grid spacing upwards at each move.
• At every move after \(P\), the token moves to the right with probability \(p\) or upwards with probability \(q\), where \(p+q=1\). The moves are independent.
• To win a round, the token must pass through \(C\) and finish at \(T\).

1. Show that the probability of winning a round is \(45p^6q^3\).[3]

It is given that \(p=\frac34\). Aisha plays one game comprising 16 rounds. Each round is independent of the others. Let \(W\) be the number of rounds that Aisha wins in one game.

2. Find the probability that Aisha wins at least 4 of the 16 rounds.[2]

3. Aisha plays 50 such games independently. Estimate the probability that the mean number of rounds she wins per game is at most 1.8. Give your answer to 3 significant figures.

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