LetX = (X1, …, Xn)be a sequence of independent, integrable[ai, bi]-valued random variables, wherea1≤ … ≤ an, b1≤ … ≤ bn. Considering the class of all such sequences, a complete comparison is made betweenM(X), the expected gain of a prophet (an observer with complete foresight), andV(X)the maximal expected gain of a gambler (an observer using only non-anticipatory stopping rules). The solution of this problem is a set in, the ‘prophet region’, which is explicitly characterized. This region yields a variety of prophet inequalities, e.g.M(X) ≤ V(X)/2ifbn= 0, bn-1= -1, an= -2andM(X) - V(X) ≤ an/2ifan> 0, bn-1= 2an, bn= 3an.