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Probability Theory

Probability theory is a branch of mathematics that investigates the probabilities associated with a random phenomenon. A random phenomenon can have several outcomes. Probability theory describes the chance of occurrence of a particular outcome by using certain formal concepts.

Probability theory makes use of some fundamentals such as sample space, probability distributions, random variables, etc. to find the likelihood of occurrence of an event. In this article, we will take a look at the definition, basics, formulas, examples, and applications of probability theory.

What is Probability Theory?

Probability theory makes the use of random variables and probability distributions to assess uncertain situations mathematically. In probability theory, the concept of probability is used to assign a numerical description to the likelihood of occurrence of an event. Probability can be defined as the number of favorable outcomes divided by the total number of possible outcomes of an event

Probability Theory Definition

Probability theory is a field of mathematics and statistics that is concerned with finding the probabilities associated with random events. There are two main approaches available to study probability theory. These are theoretical probability and experimental probability. Theoretical probability is determined on the basis of logical reasoning without conducting experiments. In contrast, experimental probability is determined on the basis of historic data by performing repeated experiments.

Probability Theory Example

Suppose the probability of obtaining a number 4 on rolling a fair dice needs to be established. The number of favorable outcomes is 1. The possible outcomes of the dice are {1, 2, 3, 4, 5, 6}. This implies that there are a total of 6 outcomes. Thus, the probability of obtaining 4 on a dice roll, using probability theory, can be computed as 1 / 6 = 0.167.

Probability Theory Basics

There are some basic terminologies associated with probability theory that aid in the understanding of this field of mathematics.

Random Experiment

A random experiment, in probability theory, can be defined as a trial that is repeated multiple times in order to get a well-defined set of possible outcomes. Tossing a coin is an example of a random experiment.

Sample Space

Sample space can be defined as the set of all possible outcomes that result from conducting a random experiment. For example, the sample space of tossing a fair coin is {heads, tails}.

Probability theory defines an event as a set of outcomes of an experiment that forms a subset of the sample space. The types of events are given as follows:

• Independent events : Events that are not affected by other events are independent events.
• Dependent events: Events that are affected by other events are known as dependent events.
• Mutually exclusive events: Events that cannot take place at the same time are mutually exclusive events.
• Equally likely events: Two or more events that have the same chance of occurring are known as equally likely events.
• Exhaustive events: An exhaustive event is one that is equal to the sample space of an experiment.

Random Variable

In probability theory, a random variable can be defined as a variable that assumes the value of all possible outcomes of an experiment. There are two types of random variables as given below.

• Discrete Random Variable: Discrete random variables can take an exact countable value such as 0, 1, 2... It can be described by the cumulative distribution function and the probability mass function.
• Continuous Random Variable: A variable that can take on an infinite number of values is known as a continuous random variable. The cumulative distribution function and probability density function are used to define the characteristics of this variable.

Probability

Probability, in probability theory, can be defined as the numerical likelihood of occurrence of an event. The probability of an event taking place will always lie between 0 and 1. This is because the number of desired outcomes can never exceed the total number of outcomes of an event. Theoretical probability and empirical probability are used in probability theory to measure the chance of an event taking place.

Conditional Probability

When the likelihood of occurrence of an event needs to be determined given that another event has already taken place, it is known as conditional probability . It is denoted as P(A | B). This represents the conditional probability of event A given that event B has already occurred.

Expectation

The expectation of a random variable, X, can be defined as the average value of the outcomes of an experiment when it is conducted multiple times. It is denoted as E[X]. It is also known as the mean of the random variable.

Variance is the measure of dispersion that shows how the distribution of a random variable varies with respect to the mean. It can be defined as the average of the squared differences from the mean of the random variable. Variance can be denoted as Var[X].

Probability Theory Distribution Function

Probability distribution or cumulative distribution function is a function that models all the possible values of an experiment along with their probabilities using a random variable. Bernoulli distribution, binomial distribution , are some examples of discrete probability distributions in probability theory. Normal distribution is an example of a continuous probability distribution.

Probability Mass Function

Probability mass function can be defined as the probability that a discrete random variable will be exactly equal to a specific value.

Probability Density Function

Probability density function is the probability that a continuous random variable will take on a set of possible values.

Probability Theory Formulas

There are many formulas in probability theory that help in calculating the various probabilities associated with events. The most important probability theory formulas are listed below.

• Theoretical probability: Number of favorable outcomes / Number of possible outcomes.
• Empirical probability : Number of times an event occurs / Total number of trials.
• Addition Rule: P(A ∪ B) = P(A) + P(B) - P(A∩B), where A and B are events.
• Complementary Rule: P(A') = 1 - P(A). P(A') denotes the probability of an event not happening.
• Independent events: P(A∩B) = P(A) ⋅ P(B)
• Conditional probability: P(A | B) = P(A∩B) / P(B)
• Bayes' Theorem: P(A | B) = P(B | A) ⋅ P(A) / P(B)
• Probability mass function: f(x) = P(X = x)
• Probability density function: p(x) = p(x) = \(\frac{\mathrm{d} F(x)}{\mathrm{d} x}\) = F'(x), where F(x) is the cumulative distribution function.
• Expectation of a continuous random variable: \(\int xf(x)dx\), where f(x) is the pdf.
• Expectation of a discrete random variable: \(\sum xp(x)\), where p(x) is the pmf.
• Variance: Var(X) = E[X 2 ] - (E[X]) 2

Applications of Probability Theory

Probability theory is used in every field to assess the risk associated with a particular decision. Some of the important applications of probability theory are listed below:

• In the finance industry, probability theory is used to create mathematical models of the stock market to predict future trends. This helps investors to invest in the least risky asset which gives the best returns.
• The consumer industry uses probability theory to reduce the probability of failure in a product's design.
• Casinos use probability theory to design a game of chance so as to make profits.

Related Articles:

• Probability Rules
• Probability and Statistics
• Geometric Distribution

Important Notes on Probability Theory

• Probability theory is a branch of mathematics that deals with the probabilities of random events.
• The concept of probability in probability theory gives the measure of the likelihood of occurrence of an event.
• The probability value will always lie between 0 and 1.
• In probability theory, all the possible outcomes of a random experiment give the sample space.
• Probability theory uses important concepts such as random variables, and cumulative distribution functions to model a random event and determine various associated probabilities.

Examples on Probability Theory

To get the sum as 8 there are 5 favorable outcomes.

[(2, 6), (6, 2), (3, 5), (5, 3), (4, 4)]

Using probability theory formulas,

Probability = Number of favorable outcomes / total number of possible outcomes.

Answer: The probability of getting the sum as 8 when two dice are rolled is 5 / 36.

Example 2: What is the probability of drawing a queen from a deck of cards?

Solution: A deck of cards has 4 suits. Each suit consists of 13 cards.

Thus, the total number of possible outcomes = (4)(13) = 52

There can be 4 queens, one belonging to each suit. Hence, the number of favorable outcomes = 4.

The card probability = 4 / 52 = 1 / 13

Answer: The probability of getting a queen from a deck of cards is 1 / 13

• Example 3: Out of 10 people, 3 bought pencils, 5 bought notebooks and 2 got both pencils and notebooks. If a customer bought a notebook what is the probability that she also bought a pencil. Solution: Using the concept of conditional probability in probability theory, P(A | B) = P(A∩B) / P(B). Let A be the event of people buying pencils and B be the event people of buying notebooks. P(A) = 3 / 10 = 0.3 P(B) = 5 / 10 = 0.5 P(A∩B) = 2 / 10 = 0.2 Substituting the values in the given formula, P(A | B) = 0.2 / 0.5 = 0.4 Answer: The probability that a customer bought a pencil given that she bought a notebook is 0.4.

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Practice Questions on Probability Theory

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FAQs on Probability Theory

What is the concept of probability theory.

Probability theory is a branch of mathematics that deals with the likelihood of occurrence of a random event. It encompasses several formal concepts related to probability such as random variables, probability theory distribution, expectation, etc.

What are the Two Types of Probabilities in Probability Theory?

The two types of probabilities in probability theory are theoretical probability and experimental probability. Theoretical probability gives the probability of what is expected to happen without conducting any experiments. Experimental probability uses repeated experiments to give the probability of an event taking place.

What are the Formulas for Probability Theory?

The main probability theory formulas are as follows:

Why is Probability Theory Used in Statistics?

Probability theory is useful in making predictions that form an important part of research. Further analysis of situations is made using statistical tools. Thus, statistics is dependent on probability theory to draw sound conclusions.

Can the Value of Probability Be Negative According to Probability Theory?

According to probability theory, the value of any probability lies between 0 and 1. 0 implies that an event does not happen and 1 denotes that the event takes place. Thus, probability cannot be negative.

What is a Random Variable in Probability Theory?

A random variable in probability theory can be defined as a variable that is used to model the probabilities of all possible outcomes of an event. A random variable can be either continuous or discrete.

What are the Applications of Probability Theory?

Probability theory has applications in almost all industrial fields. It is used to gauge and analyze the risk associated with an event and helps to make robust decisions.

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What is: Theories Of Probability

What is probability theory.

Probability theory is a branch of mathematics that deals with the analysis of random phenomena. It provides a framework for quantifying uncertainty and making predictions based on incomplete information. Probability theory is foundational to various fields, including statistics, finance, gambling, science, and artificial intelligence. By understanding the principles of probability, one can assess risks, make informed decisions, and derive meaningful insights from data.

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Basic Concepts of Probability

At the core of probability theory are several fundamental concepts, including experiments, outcomes, events, and sample spaces. An experiment is a procedure that yields one or more outcomes, while an outcome is a possible result of that experiment. An event is a specific set of outcomes, and the sample space is the collection of all possible outcomes. These concepts form the basis for calculating probabilities and understanding how likely certain events are to occur.

Types of Probability

There are several types of probability, including theoretical probability, experimental probability, and subjective probability. Theoretical probability is based on the reasoning behind probability, often derived from mathematical models. Experimental probability, on the other hand, is based on the actual results of experiments conducted. Subjective probability reflects personal beliefs or opinions about the likelihood of an event occurring, which may not be grounded in empirical evidence.

Probability Distributions

Probability distributions describe how probabilities are distributed over the values of a random variable. They can be discrete or continuous. Discrete probability distributions, such as the binomial and Poisson distributions, apply to scenarios where outcomes are countable. Continuous probability distributions, such as the normal and exponential distributions, apply to scenarios where outcomes can take on any value within a range. Understanding these distributions is crucial for data analysis and statistical inference.

Law of Large Numbers

The Law of Large Numbers is a fundamental theorem in probability theory that states that as the number of trials in an experiment increases, the empirical probability of an event will converge to its theoretical probability. This principle underpins many statistical methods and justifies the use of large samples in data analysis. It assures researchers that with enough data, their estimates will become more accurate and reliable.

Central Limit Theorem

The Central Limit Theorem (CLT) is another cornerstone of probability theory, stating that the distribution of the sum of a large number of independent and identically distributed random variables approaches a normal distribution, regardless of the original distribution of the variables. This theorem is vital for inferential statistics, as it allows statisticians to make inferences about population parameters based on sample statistics, even when the underlying distribution is not normal.

Conditional Probability

Conditional probability refers to the probability of an event occurring given that another event has already occurred. It is a critical concept in probability theory, particularly in the context of Bayesian statistics. The formula for calculating conditional probability is P(A|B) = P(A and B) / P(B), where P(A|B) is the probability of event A given event B. Understanding conditional probability is essential for analyzing dependent events and making predictions based on prior knowledge.

Bayes’ Theorem

Bayes’ Theorem is a mathematical formula that describes how to update the probability of a hypothesis based on new evidence. It combines prior probability with the likelihood of the observed data to produce a posterior probability. The theorem is expressed as P(H|E) = [P(E|H) * P(H)] / P(E), where H is the hypothesis and E is the evidence. Bayes’ Theorem is widely used in various fields, including machine learning, medical diagnosis, and risk assessment.

Applications of Probability Theory

Probability theory has numerous applications across various domains. In finance, it is used to assess risks and make investment decisions. In machine learning, probability models help in making predictions based on data. In healthcare, probability is essential for understanding the likelihood of diseases and the effectiveness of treatments. Additionally, probability plays a crucial role in quality control, insurance, and any field that involves uncertainty and decision-making.

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Often f is called the marginal distribution of X to emphasize its relation to the joint distribution of X and Y . Similarly, g ( y j ) = Σ i h ( x i , y j ) is the (marginal) distribution of Y . The random variables X and Y are defined to be independent if the events { X = x i } and { Y = y j } are independent for all i and j —i.e., if h ( x i , y j ) = f ( x i ) g ( y j ) for all i and j . The joint distribution of an arbitrary number of random variables is defined similarly.

Suppose two dice are thrown. Let X denote the sum of the numbers appearing on the two dice, and let Y denote the number of even numbers appearing. The possible values of X are 2, 3,…, 12, while the possible values of Y are 0, 1, 2. Since there are 36 possible outcomes for the two dice, the accompanying table giving the joint distribution h ( i , j ) ( i = 2, 3,…, 12; j = 0, 1, 2) and the marginal distributions f ( i ) and g ( j ) is easily computed by direct enumeration.

For more complex experiments, determination of a complete probability distribution usually requires a combination of theoretical analysis and empirical experimentation and is often very difficult. Consequently, it is desirable to describe a distribution insofar as possible by a small number of parameters that are comparatively easy to evaluate and interpret. The most important are the mean and the variance . These are both defined in terms of the “expected value” of a random variable.

If 1[A] denotes the “indicator variable” of A —i.e., a random variable equal to 1 if A occurs and equal to 0 otherwise—then E {1[ A ]} = 1 × P ( A ) + 0 × P ( A c ) = P ( A ). This shows that the concept of expectation includes that of probability as a special case.

Many probability distributions have small values of f ( x i ) associated with extreme (large or small) values of x i and larger values of f ( x i ) for intermediate x i . For example, both marginal distributions in the table are symmetrical about a midpoint that has relatively high probability, and the probability of other values decreases as one moves away from the midpoint. Insofar as a distribution f ( x i ) follows this kind of pattern, one can interpret the mean of f as a rough measure of location of the bulk of the probability distribution, because in the defining sum the values x i associated with large values of f ( x i ) more or less define the centre of the distribution. In the extreme case, the expected value of a constant random variable is just that constant.

It is also of interest to know how closely packed about its mean value a distribution is. The most important measure of concentration is the variance, denoted by Var( X ) and defined by Var( X ) = E {[ X − E ( X )] 2 }. By linearity of expectations, one has equivalently Var( X ) = E ( X 2 ) − { E ( X )} 2 . The standard deviation of X is the square root of its variance. It has a more direct interpretation than the variance because it is in the same units as X . The variance of a constant random variable is 0. Also, if c is a constant, Var( c X ) = c 2 Var( X ).

There is no general formula for the expectation of a product of random variables. If the random variables X and Y are independent, E ( X Y ) = E ( X ) E ( Y ). This can be used to show that, if X 1 ,…, X n are independent random variables, the variance of the sum X 1 +⋯+ X n is just the sum of the individual variances, Var( X 1 ) +⋯+ Var( X n ). If the X s have the same distribution and are independent, the variance of the average ( X 1 +⋯+ X n )/ n is Var( X 1 )/ n . Equivalently, the standard deviation of ( X 1 +⋯+ X n )/ n is the standard deviation of X 1 divided by Square root of √ n . This quantifies the intuitive notion that the average of repeated observations is less variable than the individual observations. More precisely, it says that the variability of the average is inversely proportional to the square root of the number of observations. This result is tremendously important in problems of statistical inference . ( See the section The law of large numbers, the central limit theorem, and the Poisson approximation .)

Properly interpreted, equation (10) is a generalization of the law of total probability.

For a simple example of the use of equation (10), recall the problem of the gambler’s ruin and let e ( x ) denote the expected duration of the game if Peter’s fortune is initially equal to x . The reasoning leading to equation (5) in conjunction with equation (10) shows that e ( x ) satisfies the equations e ( x ) = 1 + p e ( x + 1) + q e ( x − 1) for x = 1, 2,…, m − 1 with the boundary conditions e (0) = e ( m ) = 0. The solution for p ≠ 1/2 is rather complicated; for p = 1/2, e ( x ) = x ( m − x ).