# Introduction to Probability Theory

The concept of probability was born by man"s desire to know future events with certainty. From this, the study of probabilities became an often-utilized tool by the nobles to win games and other pastimes of their age. The development of these tools was a task given to court mathematicians.

With time, these mathematicians perfected these studies and found other uses for that which they had created. They continued developing the study with new methods that allowed them to maximize the use of their tools whilst diminishing the required computa- tions, and thus diminish the margins of error of their calculations.

Mathematicians throughout history have developed three different conceptual focuses to help define probability and determine the value of probability.

## The Classic Focus

This focus says that if there are x possible favorable results that entail an event A, but z possible unfavorable results which do not entail A, and all such results are either favorable or unfavorable, but not both (that is, a particular result of and event cannot be both simultaneously favorable and unfavorable), the probability of A occurring is given by

The classic focus of probability bases itself in the supposition that each result is equally likely.

This focus is called a priori because, in each case in which someone could apply it, it allows for the calculation of the value of probability before observing any sample events.

Example.

If we have 15 green stones and 9 red stones in a box, what is the probability that you will have a red stone when you bring one out of the box at random?

## The Relative Frequency Focus

This focus is also called the Empirical Focus. It determines the probability on the basis of the proportion of times that a certain event occurs throughout a certain number of observations. In this focus one does not use the notion of randomness.

Example

If you have noticed that 9 of the every 50 vehicles that pass by around a corner do not have a person wearing a seatbelt. If a transit cop were to stop someone at that corner, what is the likelihood, or probability, that he would find someone not wearing a seatbelt?

Similar to the classic focus, the empiric focus is driven by objective values of probability, and the feeling that the values indicate a long-term rate of the event"s occurrence.

## The Subjective Focus

This one says that the probability of an occurrence of an event is based on the belief of an individual, based completely on the evidence at their disposal. Under this premise, one can say that this focus is adequate only when there is an opportunity of occurrence of some event. That is, the event will occur or not occur only one time. The value of probability under this focus is determined by personal judgment.

## The Concept of Probability

One defines the calculations of probability as the conjoining of rules that allow one to determine whether a phenomenon has produced itself. This happens by founding the supposition in the calculus, statistics, and theory.

The objective of this practice is forming various experiments relating to probability, noting the results and afterwards comparing them with the theoretical results.

Probability"s Objective

The fundamental objective of probability is to demonstrate to the student the importance and utility of the Statistic Method in an economic-business role. To such an end, the student ought to learn to manage the methods and techniques most appropriate for the correct treatment and analysis of the information from the data that generates economic activity.

For that, we begin to enforce the knowledge of the student to possess the Statistic Descrip- tion on top of some new concepts related to this subject.

The Value of Probability

The smallest value which a probability can obtain is equal to 0, and such a probability indicates that the event is impossible and will never happen. Inversely, the largest value of a probability is 1, and such a probability indicates that and event will certainly occur. Then we say that if P (A) is the probability of an event A occurring, then P (At) is the probability that A does not occur. So we have

## Mutually Exclusive Events

Two or more events are mutually exclusive, or disjoint, is they cannot occur simultane- ously. That is, the occurrence of one event automatically ensures that some other mutually exclusive event will not occur.

Example.

Flipping a coin can only result in a heads or a tails, but never both. That is to say, the two events are mutually exclusive.

Two or more events are not exclusive, or not disjoint, when they can both occur simulta- neously (although not necessarily).

Example.

If we consider getting at least a white and a six in a game of dominoes, these events are not mutually exclusive because they occur at a white six.

The Rule of Addition states the following: the probability of either A or B occurring is equal to

P (A ? B) = P (A) + P (B) - P (A n B)

where P (A) and P (B) are the probabilities that A and B occur, respectively, and P (AnB) is the probability that both occur. Thus, if A and B are mutually exclusive, then P (A n

B) = 0 and then the Rule of Addition reduces to

P (A ? B) = P (A) + P (B)

Independent Events

Two or more events are independent when the occurrence or not-occurrence of one of them does not have any effect on the probability of the other"s occurrence. A typical case of independent events is "sampling with replacement", that is, when you observe a certain probabilistic system twice or more with the exact same conditions holding for the observations.

Example.

Throw a coin in the air twice. The outcome of the first flip does not effect the outcome of the second flip, and thus the events are independent.

Dependent Events

Two or more events are called dependent when the occurrence of not-occurrence of one of the events effects the probability of occurrence of the other event. When we have this case, we then employ the concepts of conditional probability in order to determine the probability of the other dependent event"s occurrence. The expression P (A|B) indicates the probability of an event A occurring given that the other event B has already occurred. It should be clear that A|B is not a fraction.

## Rules of Multiplication

These rules help determine the probability of two more more events. It"s basically equiv- alent to finding the set-intersection of the two events A and B. So, if the two events are independent, then

The Normal Probability Distribution

This is a continuous distribution of probability that is symmetric as well as unimodal, meaning it has exactly one high curve in the middle of it. This particular curve will be mentioned much in any statistical literature, especially in regards to inferential statistics, for three different reasons:

• 1. Most random processes seem to follow this distribution.

• 2. The normal probabilities can be used to approximate other distributions, such as the binomial and Poisson distributions.

• 3. By sampling processes and systems that follow this distribution, it quickly becomes clear that the samples themselves follow this distribution.

Exponential Probability Distribution

We use this distribution if we are in a Poisson process that is occurring in continuous time and space. In this case, the length of the space and time between successive events follows an exponential probability distribution. Since time and space are continuous, this distribution, like the normal distribution, is called continuous.

This distribution is best used in modeling the time between events, but it is not very good at predicting the exact moment of occurrence for some event. Instead, it is to be used to determine the chance that an event happens in some given interval of time.

Given that the Poisson process is stationary, the exponential distribution should be applied when we are interested in the time (or space) until the next event, the time between two successive events, or the time until the first occurrence of an event after some arbitrarily chosen point in time.

Example.

A maintenance department receives, on average, 5 calls an hour. Beginning with a moment randomly chosen, what is the probability of receiving a call in less than a half hour?

From the website Monografias.com

N. Ryan Karel

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