Statistics - Probability at Help With Assignment

Statistics - Probability

Probabilities express the chance of events that cannot be predicted with certainty. Even unlikely events sometimes occur. The probability of an event is viewed as a numerical measure of the chance that the event will occur. The idea is naturally relevant to situations where the outcome of an experiment or observation exhibits variation.

The term experiment is limited to studies conducted in a laboratory. Rather, it is used in a broad sense to include any operation of data collection or observation where the outcomes are subject to variation. Rolling a die, drawing a card from a shuffled deck, sampling a number of customers for an option survey and quality inspection of items from a production line are just a few examples.

An experiment is the process of observing a phenomenon that has variation in its outcomes.

The sample space associated with an experiment is the collection of all possible distinct outcomes of the experiment.

Each outcome is called an elementary outcome, a sample event, or an element of the sample space.

An event is the set of elementary outcomes possessing a designated feature.

The elementary outcomes, which together comprise the sample space, constitute the ultimate breakdown of the potential results of an experiment. For instance, in rolling a die, the elementary outcomes are points 1, 2, 3, 4, 5 and 6, which together constitute the sample space.

The probability of an event is the numerical value that represents the proportion of times the event is expected to occur when the experiment is repeated many times under identical conditions. The probability of event A is denoted by P (A)

The probability of an event is the sum of the probabilities assigned to all the elementary outcomes contained in the event. The sum of the probabilities of all the elements of S or sample space must be 1.

Probability must satisfy:

· 0 ≤ P(A) ≤ 1 for all events A

· P (A) = Σ_{all e in A} P (e)

· P (S) = Σ_{all e in S} P (e) =1

It is deduced that basic probability by reasoning from the definition that the probability of an event is the proportion of times the event is expected to occur in many repeated trails of the experiment.

The assignment of probabilities to all events in a sample space determines a probability model. In order to be a valid probability model, the probability assignment must satisfy the above conditions. Any assignment of P (e_i) to the elementary outcomes will satisfy the three conditions of probability provided these numbers are nonnegative and their sum over all the outcomes e_i in S is 1. However, the probability assigned to an event must also be in agreement with the concepts of probability as the proportion of times the event is expected to occur.

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Our previous articles on Statistics include Probability in Statistics, Using SPSS for Statistics, Hypothesis Testing, Correlation in Statistics

Probability in Statistics from HelpwithAssignment.com

Probability in Statistics from HelpwithAssignment.com

Probability is the study of random or unpredictable experiments, where it is helpful in investigating important features of random experiments. The origin of probability theory was based on the analysis of certain games of chance that was well known in 17^th century. Many branches of science and engineering have found an extensive scope of application which makes probability as one of important studies.

Sample Space and Events

In probability theory, the set of all possible outcomes of an experiment, denoted as Ω is known as Sample Space. An element in Ω is called a Sample Point or a Sample Element. An Event A is a set of particular type of outcomes of an experiment in probability theory. It is a subset of a sample space.

A simple formula explains Probability Theory

Probability (P) = Number of Events/ Total Number of outcomes

A Venn diagram of an event B is the sample space and A is an event.

By the ratio of their areas, the probability of A is approximately 0.4.

Types of Sample Spaces:

There are 3 types of Sample Spaces in probability theory are listed and explained with illustrations such as, (i) finite, (ii) countable infinite and (iii) uncountable

Illustration1. Finite: A coin is tossed in air for three times and observed the sequence of heads (H) and tails (T) that appear. Here the sample space is Ω = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} consist of eight elements and this becomes an example of finite sample space.

Illustration 2. Countable infinite: A coin is tossed and counted until the outcome appears as heads. The sample space is Ω = {1, 2, 3, 4…Infinity }. Here Infinity refers to the case when a head never appears and so the coin is tossed an infinite number of times. This is an example of Countable Infinite.

Illustration 3. Uncountable: In this example, let’s take a pencil, head first and drop it in a rectangular box, note the point on the bottom of the box that the pencil first touches. Now here Ω consists of all points on the bottom of the box.

Properties of Probability Functions:

Here are the few common properties of a probability functions which are proved by using the definition of probability function.

Theorem 1 If ø is the empty event, then P (ø) = 0

Proof: If A is an event, then A and ø are disjoint. Again A = A ø.

P (A) = P (A ø)

= P (A) + P (ø)

Hence,

P (ø) = P (A) – P (A)

= 0

Theorem 2 If A is any event in sample space Ω then P (Ă) = 1- P (A).

Proof: If A is any event, then A and Ă are disjoint and Ω = A Ă

1 = P (Ω)

= P (A U Ă)

= P (A) + P (Ă)

Hence,

P (Ă) = 1- P (A)

The two events A and B are called mutually exclusive if they are disjoint, that is, A B =ø

In other words A and B cannot occur simultaneously. A sample space, where each sample point has the same probability is called an equiprobable sample space.

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The article is in continuation with our previous articles on Probability & Statistics that include SPSS Statistics Assignment help, Statistics Assignment Help, Statistics Correlation Assignment Help, Statistics Regression Assignment Help