Suppose you have n trials of an experiment in which the probability of "success" in each trial is p. Then the probability of r successes is:
nCr*pr*(1-p)n-r for r = 0, 1, ... n.
nCr = n!/[r!*(n-r)!]
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discrete & continuous
Strictly speaking, there are no cons because they are defined for discrete variables only. The only con that I could think of is the difficulty evaluating the moments and other probabilities for some discrete distributions such as the negative binomial.
Two independent outcomes with constant probabilities.
Don't know what "this" is, but all symmetric distributions are not normal. There are many distributions, discrete and continuous that are not normal. The uniform or binomial distributions are examples of discrete symmetric distibutions that are not normal. The uniform and the beta distribution with equal parameters are examples of a continuous distribution that is not normal. The uniform distribution can be discrete or continuous.
If you're studying a subject involving or related to statistics and probability, then it will. If you're not, then it won't.