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There is no real relationship. Probabilities for the Normal distribution are extremely difficult to work out. The z-score is a method used to convert any Normal distribution into the Standard Normal distribution so that its probabilities can be looked up in tables easily. There are infinitely many types of continuous probability distributions and the Normal is just one of them.
Nearly 100%!
Random sampling is a method of selecting a sample where each member of the population has the same probability of being included in the sample. An equivalent statement is that each subset of the population, of the given size, has the same probability of being selected as any other subset of that size.
No. Probable means that a particular outcome is likely. Probability means the analytic likelihood of a particular outcome. Analysis (analytic, i.e. the method) means, for example, the evaluation of the outcomes to determine how well the experimental probability aligns with the theoretical probability.
I believe that stimulation is always inappropriate. Simulation, maybe, but that was not what the question was about.