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# Comprehensive Guide to Probability Analysis

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The word probability is derived from the word “probably” which means, likelihood or without certainty. Mathematical probability is, therefore, the measure of tendency and likelihood occurrence of an event within a given sample space (Grinstead and Snell 4). Probability estimates how an event is likely to occur within the given total outcomes. Probability of a given event ranges from 0 to 1. A calculation resulting into a probability of zero indicates no possible outcome of a given particular event. On the other hand, a calculation resulting into a probability value of one indicates 100% chances of occurrence of an event. For example, the probability that the sun will not rise today is zero, while the probability that the sun will rise is one. When tossing a coin, the probability that the head will face up is a half and also the probability that the tail will face up is also a half.

The mathematical theory of probability is dated back in 1654. Two French mathematicians Pierre de Fermat and Blaise Pascal engaged in a dispute involving gambling, which created their interest to know the probability of the particular outcomes of the game (Devlin 13). The first book on probability was thereafter published by a teacher of Leibniz in 1657. Since then, new ideas in probability have been developed and expanded to form the today’s complex and more meaningful probability.

Let P (n) denote the probability of a given event, n represent the number of ways of achievable success and N represent the total number of possible outcomes.

Then, from the definition, probability is calculated using the formula;

P (n)=n/N.

This means that the probability of picking n within a sample space N is n divided by N.

One advantage attached to the probability technique of calculation is that the probability and its statistical theory can be applied to analyze a sample of data to yield optimal estimates as well as confidential ranges for the estimates. It gives a rough idea of how a particular event is likely to occur. Probability is relatively easy to calculate. Another advantage is in its wide range of real life applications. For instance, gynecologists use this technique of calculation to advise couples about the likelihood inheritance of particular character traits, such as the inheritance of hemophilia, color blindness, and albinism.

The disadvantage of probability technique is that it gives a false sense of outcomes of how an event is likely to occur, which is not always true. For example, a gynecologist may use this technique to advise a couple on the probability of inheritance of a particular character trait, but then, the results fail to meet the couple’s interest. Another disadvantage of probability techniques is that, calculations involving multiple outcomes are hard to perform.

Question 1: Two Main Types of Probability Interpretations with Conflicting Views

The first Major category of probability interpretation is the Subjective probability.   Subjective probability involves more of a personal belief than the actual outcome. It relies on past personal experience on the similar outcome. If for example, the probability of a particular event has been high in the past outcome, then one may assume that it is going to be high in the preceding outcome. The second category of probability interpretations is the Objective probability. This is more of a practical than a theory type of probability. For instance, picking a playing card out of a deck, or rolling a dice results into the objective probability.

Question 2: Calculation of Probability of an Adult American Has Never Been Tested

Consider the data in the table below;

To determine this probability,

Let p denote the probability that a randomly selected American adult has never been tested,

p(NT)  denote the total number of American adults never tested,

p(T) denote the total number of American adults  tested,

Then we use the formula;

P=p(NT)/{p(NT)+p(T)}.

Now using the data in the total rows, we have;

p(NT)=134,767

P(T)=77,789

p(NT)+ P(T)= 212,556

p(NT)/{p(NT)+p(T)}=134,767÷(134,767+77,789)= 134,767÷212,556=0.63

Therefore, p=0.63

Question 3: Proportion of Americans aged between 18 and 44 who have never undergone the HIV test

To calculate the proportion, we apply the given formula;

P= p(NT)/{p(NT)+p(T)}

p(T)}=50,080

p(NT)=56,405

{p(NT)+p(T)}=106,485

Now, p=p(NT)/{p(NT)+p(T)}=56,405÷(56,405+50,080)=0.53

Therefore, p=0.53

Probability is a mathematical tool that originated from a humble background. Today, the probability technique has evolved to become one of the commonly used mathematical calculation technique across the world of mathematics. Probability has a wide range of applications. It is applicable in many fields such as, statistical companies, in the field of medicine, gambling companies, in the field of engineering, industrial chemist, among many others. Our daily activities also involve a wide range of probability, whether we are aware or not.

Works Cited

Devlin, Keith J. The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter That Made the World Modern. New York: Basic Books, 2008. Print.

Grinstead, Charles, and J L. Snell. Introduction to Probability. Place of publication not identified: publisher not identified, 1997. Internet resource.

September 25, 2023
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