In all, at least well-known probabilities, the parameters are not related to mean and variance. In other words, usually we provide a specification on how a single events occur (e.g., p in Bernoulli trial, frequency in Poisson / exponential). Then, we can calculate the expected value and variance. In contrast, in Normal distribution we provide mean (which is expected value) and variance (standard deviation). So, this looks as though this is not an information about single events.
Why is this so different for Normal distribution?