Reading Z off a rate-decision market

The central bank meets tomorrow and will cut, hold or hike. Your model gives real-world probabilities $0.2$, $0.5$ and $0.3$. Digital contracts paying $1 in exactly one outcome trade at 0.30 (cut), 0.45 (hold) and 0.25 (hike). Rates are zero over one day.

(a) Find the risk-neutral probabilities and $Z = d\mathbb Q/d\mathbb P$ in each outcome. Check that $E[Z] = 1$.

(b) A position pays $X = 6$ on a cut, $1$ on a hold and $-5$ on a hike. Price it using $E[ZX]$, and find its real expected payoff.

(c) Which outcome does the market treat as the bad state, and how can you tell from $Z$?