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We study the first passage statistics to adsorbing boundaries of a Brownianmotion in bounded two-dimensional domains of different shapes andconfigurations of the adsorbing and reflecting boundaries. From extensivenumerical analysis we obtain the probability P(\omega) distribution of therandom variable \omega=\tau_1/(\tau_1+\tau_2), which is a measure for howsimilar the first passage times \tau_1 and \tau_2 are of two independentrealisations of a Brownian walk starting at the same location. We construct achart for each domain, determining whether P(\omega) represents a unimodal,bell-shaped form, or a bimodal, M-shaped behaviour. While in the former casethe mean first passage time (MFPT) is a valid characteristic of the firstpassage behaviour, in the latter case it is an insufficient measure for theprocess. Strikingly we find a distinct turnover between the two modes ofP(\omega), characteristic for the domain shape and the respective location ofabsorbing and reflective boundaries. Our results demonstrate that largefluctuations of the first passage times may occur frequently in two-dimensionaldomains, rendering quite vague the general use of the MFPT as a robust measureof the actual behaviour even in bounded domains, in which all moments of thefirst passage distribution exist.

First passages in bounded domains: When is the mean first passage time meaningful?