Dependent probabilities calculated in the probability and contingency fields. Presentation of the concepts of multiplicative, additive and conditional probability

$$p(B|A)$$

and above all Total probability within Bayes' theorem, where

$$p(A|B)=\frac{p(B|A)⋅pA}{pB}.$$

Important works on the basis of probability theory can be found in Cardano (1545), Arnauld and Nicole (1662), de Moivre (1711), Bernoulli (1713), Bayes and Price (1763) also Hacking (1975) or Hald (1990).

A deeper insight into probability theory and probability calculation give e.g. Bortz and Schuster (2010) or Harney (2016).

Arnauld, A., & Nicole, P. (1662). La logique ou L’art de penser. 1st ed. A Paris: Chez Charles Savreux, au pied de la Tour de Nostre Dame. https://gallica.bnf.fr/ark:/12148/bpt6k574432.image

Bortz, J., & Schuster, C. (2010). Statistik Für Human- Und Sozialwissenschaftler: Limitierte Sonderausgabe. 7th ed. Springer-Lehrbuch. Berlin, Heidelberg: Springer. https://doi.org/10.1007/978-3-642-12770-0

Bayes, T., & Price, R. (1763). An Essay Towards Solving a Problem in the Doctrine of Chances. By the Late Rev. Mr. Bayes, f. R. S. Communicated by Mr. Price, in a Letter to John Canton, a. M. F. R. s. Philosophical Transactions (1683-1775), 53, 370–418. http://www.jstor.org/stable/105741

Bernoulli, J. (1713). Ars conjectandi, opus posthumum. Accedit Tractatus de seriebus infinitis, et epistola gallicé scripta de ludo pilae reticularis. Basileae: Impensis Thurnisiorum, Fratrum. https://www.e-rara.ch/zut/doi/10.3931/e-rara-9001

Cardano, G. (1545). HIERONYMI CARDANI, ARTIS MAGNÆ, SIVE DE REGVLIS ALGEBRAICIS, LIBER VNVS. S. P. D: ANDREÆ OSIANDRO viro eruditiss. https://web.archive.org/web/20220201093634/http://www.filosofia.unimi.it/cardano/testi/operaomnia/vol_4_s_4.pdf

de Moivre, A. (1711). De mensura sortis, seu, de probabilitate eventuum in ludis a casu fortuito pendentibus. Philosophical Transactions of the Royal Society of London, 27(329), 213–64. https://doi.org/10.1098/rstl.1710.0018

Hacking, I. (1975). The Emergence of Probability: A Philosophical Study of Early Ideas About Probability, Induction and Statistical Inference. Cambridge University Press. https://philpapers.org/rec/HACTEO-8

Hald, A. (1990). History of Probability and Statistics and Their Applications before 1750. New York: Wiley Series in Probability; Statistics, Wiley-Interscience. https://onlinelibrary.wiley.com/doi/book/10.1002/0471725161

Harney, H. L. (2016). Bayes’ Theorem. In Bayesian Inference: Data Evaluation and Decisions, 11–25. Springer International Publishing. https://doi.org/10.1007/978-3-319-41644-1_2