Focusing on takeover bids for which the outcome can be predicted in advance with certainty, Grossman and Hart established the proposition, which subsequent work accepted, that successful bids must be made at or above the expected value of minority shares. This proposition provided the basis for Grossman and Hart's identification of a free-rider problem and became a major premise for the analysis of takeovers. This paper shows that this important proposition does not always hold once we drop the assumption that the only successful bids are those whose success could have been predicted with certainty. In particular, it is shown that any unconditional bid that is below the expected value of minority shares but above the independent target's per share value will succeed with a positive probability, that the bidder's expected payoff from such a bid (not counting the transaction costs of making the bid) is always positive, and that bidders might elect to make such bids. These results have implications for the nature of the free-rider problem and for the operation of takeovers; in particular, it is shown that, when a raider can increase the value of a target's assets, the raider might elect to bid even if no dilution of minority shares is possible and it holds no initial stake in the target.