 The rough idea is to exploit the richness of the space of prizes, which ensures that for any gamble \(g\) that yields better prize \(b\) if \(E\) is true and worse prize \(w\) if \(E\) is false, the agent is indifferent between \(g\) and some middling prize \(m\).  This means that \(EU(g) = EU(m)\).  Using some algebra, plus the fact that \(EU(g) = P(E)U(b) + (1-P(E))U(w)\), Ramsey shows that