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Central theorems of probability theory and their impact on probabilistic intuitions

Manfred Borovcnik

University of Alpen-Adria, Klagenfurt

Abstract

Probability is a difficult concept and there are many misleading intuitions. Unlike in geometry our perception has not been trained to improve our ideas as probability is not a physical property in the real world. Yet it is often equated to the relative frequencies of an event in a series of repeated experiments. In fact, there is a relation between the two concepts (if only such an experiment could be repeated under the same conditions) – though this relation is a bit more complicated. Some statisticians therefore prefer to speak of probability as a metaphor to speak about a random situation, or they would state that probability is a virtual concept (like the Internet or computer games are virtual worlds).

Mathematically, two groups of central theorems regulate what probability is and how we can interpret it. The one group of theorems is the laws of large numbers; the other is the group of central limit theorems. The first justifies that we interpret probability in terms of relative frequencies. The basic law of large numbers is usually summarized as: the relative frequencies “converge” to the (possibly) unknown probability of the event under scrutiny. The second explains why we can describe the variation of a random variable by a normal distribution in quite a few cases (and becomes eminently important in statistical inference). The simplest case has become famous in the history of probability as the law of errors, which is a thought experiment: if a measurement error can be explained by a sum of elementary errors (each of them is not observable) then the resulting error (that can be observed) should follow a normal distribution.

The simplifying statement for the law of large numbers is simply wrong and misleading but is has a true kernel. We could look more precisely at the mathematical theorem but this requires quite a lot of mathematical arguments. The question is how to develop scenarios and formal signs (with accompanying pictures behind) that we can teach the topic and communicate its relevance, shape intuitions that comply with the mathematical background, and “revise” intuitions that are at least not helpful (if not wrong). How can we explain at an intuitive level, in which sense and under which conditions the relative frequencies do converge to the underlying probability? The simplifying statement for the central limit theorem is simply wrong as the sum of the elementary errors cannot converge as it tends to get larger and larger if we add more elementary errors (even with an increasing variability). Again, the teaching challenge is to investigate various situations and observe a kind of divergence or convergence. A further challenge is to clarify the kind of convergence to the normal distribution and design situations where such knowledge would be helpful.

We will use simulations of random experiments and didactical animations of binomial distributions and investigate the “data” from various perspectives to support feasible ideas about the Central Limit theorem, which will help to understand how the concept of probability may be used to extract information from data.

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