Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 12 / b / iii / Solution

The equality must be a first return time: an order starting $GR$ or $RG$ would already have stopped at draw two. The only successful four-draw orders are therefore $GGRR$ and $RRGG$. Their probabilities under sampling without replacement add toHere $N=g+r$. Counting all balanced four-draw orders would include games that have already stopped.