What mathematicians call probability is the mathematical theory we use to describe and quantify uncertainty. In a larger context, (see probability interpretations) the word probability is used with other concerns in mind. Uncertainty can be due to our ignorance, deliberate mixing or shuffling, or due to the essential randomness of Nature. In any case, we measure the uncertainty of events on a scale from zero (impossible events) to one (certain events or no uncertainty).
Probability axioms form the basis for mathematical probability theory. Calculation of probabilities can often be determined using combinatorics or by applying the axioms directly. Probability applications include even more than Statistics, which is usually based on the idea of probability distributions.
Probability theory plays a critical role in the development of statistical theory. Statistics is a branch of applied mathematics which includes planning, summarizing, and interpreting uncertain observations. We describe our knowledge (and ignorance) mathematically and attempt to learn more from whatever we can observe. This requires us to
- plan our observations to control their variability (planning statistical research),
- summarize a collection of observations to feature their communality by suppressing details (summarizing statistical data), and
- reach consensus about what the observations tell us about the world we observe (interpreting statistical data).
In some forms of descriptive statistics, notably data mining the second and third steps become so prominent that the first step (planning) appears to recede in importance. In these disciplines, the data is often collected outside the control of the person doing the analysis, and the result of the analysis may be more an operational model than a consensus report about the world.
- Psychological statistics
- Business statistics
- Economic statistics
- Engineering statistics
- Social statistics (for all the social sciences)
- statistical theory -- applied statistics -- planning statistical research -- summarizing statistical data -- interpreting statistical data -- information theory
Some valuable resources on the Web include: