We study the first-passage time distribution (FPTD) F(t_{f}|x_{0},L) for a freely diffusing particle starting at x_{0} in one dimension, to a target located at L, averaged over the initial position x_{0} drawn from a normalized distribution (1/σ)g(x_{0}/σ) of finite width σ. We show the averaged FPTD undergoes a sharp dynamical phase transition from a two-peak structure for b=L/σ>b_{c} to a single-peak structure for b<b_{c}. This transition is generated by the competition of two characteristic timescales σ^{2}/D and L^{2}/D, where D is the diffusion coefficient. A very good agreement is found between theoretical predictions and experimental results obtained with a Brownian bead whose diffusion is initialized by an optical trap which determines the initial distribution g(x_{0}/σ). We show that this transition is robust: It is present for all initial conditions with a finite σ, in all dimensions, and also exists for more general stochastic processes going beyond free diffusion.