After a general discussion of diffusion in a potential field having two minima, an explicit solution is obtained for a special form of the potential. The potential is symmetric, but the diffusing particle is initially at an arbitrary point near the potential maximum. Various suggested approximation schemes are tested, with the following conclusions, (i) In linear approximation around the maximum the probability distribution is Gaussian. A two-peaked distribution emerges only through nonlinear terms, (ii) The chances for the particle to go to the right or the left valley cannot be found from the linear approximation. Matching the linear approximation with a macroscopic description far away from the maximum is therefore wrong, (iii) Kramers' treatment of the escape across a potential barrier yields a practically exact result for this model.