The odds theorem gives a unified answer to a class of stopping problems on sequences of independent indicator functions. It can be applied to natural stopping problems such as, for example, the secretary problem, the group-interview problem, the last-peak problem, but also to many other simple problems of games, betting or investment. The success probability of the optimal rule is known to be larger than \(R e^{-R}\), where \(R\) defined in the theorem satisfies \(R \geq 1\) in the more interesting case. This result is strenghened by showing that \(1/e\) is then a lower bound. This best possible uniform lower bound extends to the general setting of the odds theorem.