Systematic trading contingent on observed prices by agents uninformed about fundamentals has long been considered at odds with efficient markets populated by rational agents. In this paper we show that price-contingent trading is the equilibrium strategy of rational agents in efficient markets in which there is uncertainty about whether a large trader is informed. In this environment, knowing his own type and past trades (or lack of them) will be enough for a large trader to retrieve some private information about the fundamental indirectly even if he does not observe fundamental information directly. Such trader pursues price-contingent trading which remains profitable in a (semi-strong) efficient market. Our results generalize to a large variety of distributional assumptions. We then provide conditions under which price-contingent trading is positive-feedback or contrarian. On average both positive-feedback and contrarian trading help prices converge faster to fundamentals, although they can occasionally trigger divergence.