Abstract
We begin our discussion of probabilities with the definition of relative frequency, because this notion is very concrete and probabilities are, in a sense, idealizations of relative frequencies.
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Notes
- 1.
If S is a finite set, then the collection \(\mathcal{F}\) of events is taken to be the collection of all subsets of S. If S is infinite, then \(\mathcal{F}\) must be a so-called sigma-field, which we do not discuss here.
- 2.
It is customary to omit the braces in writing the probabilities of the elementary events, such as writing P(s 1) instead of the correct, but clumsy, P\((\left \{s_{1}\right \}).\)
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- 4.
Note that, terminology notwithstanding, it is the events with their probabilities that are here defined to be independent, not just the events themselves.
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Named after one of the founders of the theory of probability, Jacob Bernoulli (1654–1705), the most prominent member of a Swiss family of at least six famous mathematicians.
- 6.
Note, however, that such a multiplication rule does hold for expected values. In this case, the expected number of double sixes in n throws is n times the expected number in one throw, as we shall see in Section 6.1
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Because of this theorem, a few authors use the notation P\(_{B}\left (A\right )\) for P\(\left (A\vert B\right )\) to emphasize the fact that P B is a probability measure on S and in P\(\left (A\vert B\right )\) we do not have a function of a conditional event A | B but a function of A. In other words, P\(\left (A\vert B\right ) =\) (the probability of A) given B, and not the probability of (A given B). Conditional events have been defined but have not gained popularity.
- 9.
We usually omit the braces or union signs around compound events when there are already parentheses there, and separate the components with commas. Thus we write P\(\left (CC,C\overline{C},\overline{C}C\right )\) rather than P\(\left (\{CC,C\overline{C},\overline{C}C\}\right )\) or P\(\left (CC \cup C\overline{C} \cup \overline{C}C\right ).\)
- 10.
Tversky’s Legacy Revisited, by Keith Devlin,
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Schay, G. (2016). Probabilities. In: Introduction to Probability with Statistical Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-30620-9_4
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DOI: https://doi.org/10.1007/978-3-319-30620-9_4
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