  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
Schr\"odinger 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.
