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Probability Theory

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Statistics for Data Scientists

Part of the book series: Undergraduate Topics in Computer Science ((UTICS))

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Abstract

Statistics is a science that is concerned with principles, methods, and techniques for collecting, processing, analyzing, presenting, and interpreting (numerical) data. Statistics can be divided roughly into descriptive statistics (Chap. 1) and inferential statistics (Chap. 2), as we have already suggested. Descriptive statistics summarizes and visualizes the observed data. It is usually not very difficult, but it forms an essential part of reporting (scientific) results. Inferential statistics tries to draw conclusions from the data that would hold true for part or the whole of the population from which the data is collected. The theory of probability, which is the topic of the next two theoretical chapters, makes it possible to connect the two disciplines of descriptive and inferential statistics. We have already encountered some ideas from probability theory in the previous chapter. To start with, we discussed the probability of selecting a specific sample \(\pi _k\) and we briefly defined the notion of probability based on the throwing of a dice. In this chapter we work out these ideas more formally and discuss the probabilities of events; we define probabilities and discuss how to calculate with probabilities. In the previous chapter, when discussing bias, we have also encountered the expected population parameter \(\mathbb {E}(T)\), but we have not yet detailed what expectations are exactly; this is something we cover in Chap. 4.

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Notes

  1. 1.

    It should be noted here that a probability of zero does not necessarily mean that the event will never occur. This seems contradictory, but we will explain this later. On the other hand, if the event can never occur, the probability is zero.

  2. 2.

    Using definition Eq. (3.3) we can write \(\Pr (A\cap B)\) as \(\Pr (A|B)\Pr (B)\), as we did in Table 3.1, but also as \(\Pr (B|A)\Pr (A)\). Which one to use mostly depends on the practical situation. In Table 3.1 we could have used \(\Pr (B|A)\Pr (A)\) as well.

  3. 3.

    If, in this case, the population size(s) were known, we could calculate weighted averages to estimate the population parameters as we did in Chap. 2.

  4. 4.

    Note that Simpson’s Paradox, and its solutions, are still heavily debated (see, Armistead 2014 for examples).

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Authors and Affiliations

  1. Tilburg University, Tilburg, Noord-Brabant, The Netherlands

    Maurits Kaptein

  2. Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, Noord-Brabant, The Netherlands

    Edwin van den Heuvel

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  1. Maurits Kaptein
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  2. Edwin van den Heuvel
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Correspondence to Maurits Kaptein .

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Kaptein, M., van den Heuvel, E. (2022). Probability Theory. In: Statistics for Data Scientists . Undergraduate Topics in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-030-10531-0_3

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  • DOI: https://doi.org/10.1007/978-3-030-10531-0_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-10530-3

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