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

After one draw the two counts are either $(1,0)$ or $(0,1)$ and cannot agree. ThereforeThis is the elementary parity restriction behind the first return time interpretation of this game.

Success after two draws occurs on the two mutually exclusive events $GR$ and $RG$. With $N=g+r$, sampling without replacement gives

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.