Introduction to the Theory of Probability

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* Introduction to the Theory of Probability

* Conceptual Approaches

* Concept of Probability

* Objectives

* Value of Probability

* Mutually Exclusive Events and inclusive

* Adding Rules

* Independent Events

* Dependent Events

* Multiplication Rules

* Normal Probability Distribution

* Exponential probability distribution

*

INTRODUCTION TO PROBABILITY THEORY

The concept of probability is born with man’s desire to know with certainty the future events. That is why the study of probability arises as a tool used by nobles to win the games and pastimes of the era. The development of these tools was assigned to the mathematicians of the court.

Over time these mathematical techniques were refined and found other uses very different for which they were created. Today we continued the study of new methodologies to maximize the use of computers in the study of probability decreasing, thus the margin of error in the calculations.

Throughout history, people have developed three different conceptual approaches to define the likelihood and probability values determine:

The classical approach

He says if there is x possible outcomes favorable to the occurrence of an event A yz possible unfavorable results to the occurrence of A, and all results are equally possible and mutually exclusive (they can occur both at the same time), then the probability of A occurring is:

The classical approach is based on the probability of each outcome assumption is also possible.

This approach is called a priori approach because it allows (if it can be applied) to calculate the probability value observed before any sampling event.

Example:

If we have 15 stones in a box green and 9 red stones. The probability of drawing a red stone in a bid is:

The relative frequency approach

Also called empirical approach, determines the probability based on the proportion of times an event occurs in a number of positive comments. This approach uses the assumption that no prior randomness. For the determination of the probability values is based on observation and data collection.

Example:

It has been found that 9 out of 50 vehicles passing through a corner having no seat belt. If a traffic policeman stands on the same corner a round any What will the probability of stopping a vehicle without a seat belt

Both the classical approach as the empirical approach targets leading to probability values, in the sense that the probability values indicating the long term relative rate of occurrence of the event.

The subjective approach

Says that the probability of occurrence of an event is the degree of belief of an individual that an event will occur, based on all available evidence. Under this premise, we can say that this approach is appropriate when there is only one chance of occurrence of the event. That is, the event that occur or not occur once. The probability value under this approach is a judgment call.

Concept of Probability

It is defined as the probability calculation rule set for determining whether a phenomenon has occurred, basing the assumption in the calculation, statistics or theory.

The purpose of this practice is likely to perform several experiments, record the results and then compare them with the theoretical results.

Objectives of Probabilities

The fundamental objective of probability, is to show students the importance and utility of the statistical method in the economic-business. To this end, students will learn how to handle the methods and techniques suitable for the proper treatment and analysis of information provided by the data it generates economic activity.

This will begin consolidating the knowledge that the student already has a statistics, plus some new concepts related to this topic.

The probability value

The value can be smaller than the probability of occurrence of an event is equal to 0, which indicates that the event is impossible, and the highest value is 1, which indicates that the event will certainly occur. So if we say that P (A) is the probability of occurrence of an event A and P (A ‘) the probability of non-occurrence of A, we have:

Events mutually exclusive and nonexclusive events

Two or more events are mutually exclusive or disjoint, if they can not occur simultaneously. That is, the occurrence of an event automatically prevents the occurrence of another event (or events).

Example:

By flipping a coin can only be that getting heads or seal but not both at the same time, this means that these events are exclusive.

Two or more events are not mutually exclusive, or sets, it is possible that both occur. This should not necessarily indicate that these events occur simultaneously.

Example:

If we believe in a domino game score at least one white and one six, these events are not mutually exclusive because they may leave the white six.

Adding Rules

The addition rule states: the probability of occurrence of at least two events A and B is equal to:

P (A or B) = P (A) UP (B) = P (A) + P (B) if A and B are mutually exclusive

P (A or B) = P (A) + P (B) – P (A and B) if A and B are not mutually exclusive

Where: P (A) = probability of occurrence of event A

P (B) = probability of occurrence of the event B

P (A and B) = probability of simultaneous occurrence of events A and B

Independent Events

Two or more events are independent if the occurrence or non-occurrence of one event has no effect on the probability of occurrence of another event (or events). A typical case of independent events is sampling with replacement, ie, when the sample is taken back again to the town where it was obtained.

Example:

tossing a coin twice are independent events that the outcome of the first event has no effect on the probability of occurrence of effective heads or tails, on the second pitch.

Dependent Events

Two or more events are dependent if the occurrence or non-occurrence of one affects the probability of occurrence of the other (or others). When we this case, then employ the concept of conditional probability to the probability of call related event. The expression P (A | B) indicates the probability of occurrence of event A if the event B has already occurred.

It should be clear that A | B is not a fraction.

P (A | B) = P (A and B) / P (B) or P (B | A) = P (A and B) / P (A)

Multiplication Rules

They relate to the determination of the joint occurrence of two or more events. I.e. the intersection between the sets of possible values of A and B values, this means that the probability of the joint events A and B is:

P (A and B) = P (AB) = P (A) P (B) if A and B are independent

P (A and B) = P (AB) = P (A) P (B | A) if A and B are dependent

P (A and B) = P (AB) = P (B) P (A | B) if A and B are dependent

Normal Probability Distribution

It is a continuous probability distribution is symmetrical both as mesokurtic. The curve representing the normal probability distribution is generally described as bell shaped. This distribution is important in statistical inference for three different reasons:

* It is known that measurements produced in many random processes follow this distribution.

* The normal probability can generally be used to approximate other probability distributions such as binomial and Poisson distributions.

* The statistical distributions such as sample mean and sample rate, often follow normal distribution, regardless of the distribution of the population

The values of the parameters of the normal probability distribution are = 0 and = 1. Any set of values X may become normally distributed standard normal z by the formula:

Enabling the use of the table of area ratios and makes the use of the equation of the probability density function of any given normal distribution.

To approximate the discrete binomial distributions and Poisson do:

Exponential probability distribution

If in the context of a Poisson process events or successes in a continuum of time and space. Then the length of the space or time between successive events follows an exponential probability distribution. Since time and space are a continuum, this is a continuous distribution.

If this type of distribution is worth asking what is the probability that the first service request is made exactly in about a minute. Rather we must assign a range within which the event can occur, wondering, what is the probability that the first order occurs in the next minute.

Since the Poisson process is stationary, the exponential distribution applies either when we are interested in time (or space) to the first event, the time between two successive events, or the time until the first event occurs after any randomly selected point.

Where is the average number of occurrences for the interval of interest, the exponential probability that the first event occurs within the designated interval of time or space is.

P (T

Thus the exponential probability that the first event occurring within the interval not designated time or space is:

P (T> t) = e –

Example:

A maintenance department receives an average of 5 calls per hour. Beginning at a randomly selected time, the probability that a call arrives within half an hour is:

Average 5 per hour, as the interval is half an hour we have = 2.5 / half hour.

P (T <30 min.) = 1 - e -5 = 1-.08208 = .91792

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