In the continuous probability space (e.g. a real number picked uniformly at random between 0 and 1) single outcomes have infinitesimally small probabilities (i.e. probability of said number being exactly 0.5 is 0%). Therefore it only makes sense to assign probabilities to subspaces of outcomes (e.g. said number being between 0.45 and 0.55, where the probability is 10%).
To achieve this, continuous probability distributions are defined through probability density functions, as opposed to probability mass functions in the discrete case. Then the average prob. density of a subspace is multiplied by its size to obtain the probability of an event from the subspace happening.
