The single-particle density is the most basic quantity that can be calculated from a given many-body wave function. It provides the probability to find a particle at a given position when the average over many realizations of an experiment is taken. However, the outcome of single experimental shots of ultracold atom experiments is determined by the $N$-particle probability density. This difference can lead to surprising results. For example, independent Bose-Einstein condensates (BECs) with definite particle numbers form interference fringes even though no fringes would be expected based on the single-particle density [1-4]. By drawing random deviates from the $N$-particle probability density single experimental shots can be simulated from first principles [1, 3, 5]. However, obtaining expressions for the $N$-particle probability density of realistic time-dependent many-body systems has so far been elusive. Here, we show how single experimental shots of general ultracold bosonic systems can be simulated based on numerical solutions of the many-body Schrodinger equation. We show how full counting distributions of observables involving any number of particles can be obtained and how correlation functions of any order can be evaluated. As examples we show the appearance of interference fringes in interacting independent BECs, fluctuations in the collisions of strongly attractive BECs, the appearance of randomly fluctuating vortices in rotating systems and the center of mass fluctuations of attractive BECs in a harmonic trap. The method described is broadly applicable to bosonic many-body systems whose phenomenology is driven by information beyond what is typically available in low-order correlation functions.
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