**Probability Theory**

The word probability is derived from the Latin word *probabilis*, which means commendable. In Latin, the word closely means the same as proof, which indicates the extent to which a phenomenon is “provable” (Mike, 2020). Probability is a mathematical branch that deals with analyzing and predicting the likelihood of an event occurring. It deals with analyzing random phenomena which has a number of possible outcomes. The outcomes of such events are determined by chance. Probability has been applied in different contexts, including ordinary conversations and interpretation of simple events, such as in games of cards, coins, roulette wheels, and dice (Lee, 2008). Scientists also use the theory of probability to explain their various theories, such as induction in science. However, for all the contexts, the overarching concept remains the measure of chance.

Early instances of probability use date back to the 8^{th} and 13^{th} centuries when the early Arab Mathematicians incorporated the concept to predict the possibility of Arabic words with or without vowels through permutation and combination (Broemeling, 2011). The history of modern probability theory can be traced back to the 1650s by two French mathematicians, Pascal Blaise and Fermat de Pierre, as a result of concern over the issue of points in gambling games. The major question that arose was how to divide the stakes between two players who leave a game before its completion. To solve the matter, Pascal suggested the expression p=0.5 to come up with the coefficients that defined the probability (Rowbottom, 2015). In these games, the calculated likelihood of equal outcomes led to the actual definition of probability.

**Advantages of Probability Theory**

- The calculations are useful for the quantitative analysis of data regarding human activities
- Helps describe complex systems using partial information regarding their state.
- Takes lesser time to establish the possible range

**Disadvantages of Probability Theory**

- Some probability calculations can be redundant and monotonous
- Some calculations, such as Bayesian calculations, can be lengthy and require technical knowledge.

**Two Interpretations of Probability According to the Two Adherents**

Probability can be interpreted under two broad categories, *physical and evidential* probability. Adherents of physical interpretation include von Mises, Venn, and Reichenbach under the frequentist account and Miller, Giere, Fetzer, and Popper under the propensity perspective (Siegmund, 2018). These explain the physical systems, such as rolling dice, rolling wheels, and dice rolls, which occur at relative frequencies. On the other hand, *evidential probability* adherents include Cox and Ramsey for epistemic interpretation and the likes of Carnap and Keynes for logical interpretation (Siegmund, 2018). *Evidential* interpretation relies on the available evidence to support the plausibility of a phenomenon.

**The Probability That A Randomly Selected American Adult Has Never Been Tested**

=0.6340

**The Proportion of 18- to 44-Year-Old Americans That Have Never Been Tested for HIV**

=0.5297

**References**

Broemeling, L. D. (1 November 2011). “An Account of Early Statistical Inference in Arab Cryptology”. The American Statistician. 65 (4): 255–257. doi:10.1198/tas.2011.10191.

Lee, P. (2008). History of probability theory. In Rudas, T. *Handbook of probability: Theory and applications* (pp. 3-14). Thousand Oaks, CA: SAGE Publications, Inc. doi: 10.4135/9781452226620

Mike, V. (2020). “Probability: History, Interpretation, and Application.” Encyclopedia of Science, Technology, and Ethics. Retrieved October 16, 2020, from Encyclopedia.com: https://www.encyclopedia.com/science/encyclopedias-almanacs-transcripts-and-maps/probability-history-interpretation-and-application

Rowbottom, D. (2015). *Probability*. Cambridge: Polity. ISBN 978-0745652573.

Siegmund, D. (2018). “Probability theory” Encyclopedia Britannica. https://www.britannica.com/science/probability-theory

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**Question**

**Probability Theory**

Suppose that a 2012 National Health Interview Survey gives the number of adults in the United States which gives the number of adults in the United States (reported in thousands) classified by their age group, and whether or not respondents have ever been tested for HIV. Here are the data:

Age Group | Tested | Never Tested |

18–44 years | 50,080 | 56,405 |

45–64 years | 23,768 | 48,537 |

65–74 years | 2,694 | 15,162 |

75 years and older | 1,247 | 14,663 |

Total | 77,789 | 134,767 |

Discuss probability. What is its history? What is the theory of probability? How is it calculated? What are the advantages and disadvantages of using this technique?

- Identify and discuss the two major categories of probability interpretations, whose adherents possess conflicting views about the fundamental nature of probability.
- Based on this survey, what is the probability that a randomly selected American adult has never been tested? Show your work. Hint: using the data in the two total rows, this would be calculated as p (NT) /( p (NT) + p (T)), where p is probability.
- What proportion of 18- to 44-year-old Americans have never been tested for HIV? Hint: using the values in the 18–44 cells, this would be calculated as p (NT) / ( p (NT) + p (T)), where p is probability. Show your work.

Submit your (2-3 pages) paper by the end of this module.