All the time. Statistic is based on the application of probability theory!
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Probability theory and distributive theory.
Yes; the p value used in hypothesis testing is probability. See the related link.
Probability is related to statistics in a direct manner. When one is doing a research for statistics, probability has to be used especially in sampling a small region.
Statistics consists of Descriptive Statistics,Probability theory,Distribution theory,Quality Control, Design of Experiments, Reliability, Operations Research, Queuing theory, Inventory control,Measure theory, Sampling theory, Statistical inference, Analysis.
One subject they're important in is physics. Statistics play such a big role in thermal dynamics that it is often referred to as statistical mechanics. Also, probability theory uses statistics as its base and quantum mechanics is all about probability.