Yes.
In applying inferential statistical methods the assumption is made that at least some of the variation observed in the data is due to randomness and that the laws of probability can be applied to take this into account in drawing conclusions from the data.
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Probability is the theoretical basis that underpins statistics.
Yes; the p value used in hypothesis testing is probability. See the related link.
Why are measures of variability essential to inferential statistics?
There are two types of statistics. One is called descriptive statistics and the other is inferential statistics. Descriptive statistics is when you use numbers. Inferential statistics is when you draw conclusions or make predictions.
Descriptive statistics are meant to describe the situation such as the average or the range. Inferential statistics is used to differentiate between a couple of groups.