# Probability theory

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**Probability theory** is a branch of mathematics that deals with calculating the likelihood of an event occurring. It is used to quantify the uncertainty associated with a given event, and can help to make decisions based on the likelihood of an event occurring. Probability theory is based on the idea that all possible outcomes of an experiment can be assigned a probability value, which is a number between 0 and 1 that indicates how likely it is for that particular outcome to occur. Probability theory is used in many areas, including decision making, game theory, and statistics.

The basic principles of probability theory include the following:

**The Law of Large Numbers**: This states that as the number of trials of an experiment increases, the results will approach the expected probability of the event.**The Addition Rule**: This states that the probability of two or more events occurring is equal to the sum of the probabilities of the individual events.**The Multiplication Rule**: This states that the probability of two or more events occurring simultaneously is equal to the product of the probabilities of the individual events.**Bayes' Theorem**: This theorem states that the conditional probability of an event occurring is equal to the product of the probability of the event and the probability of the event occurring given the other events.

## Example of Probability theory

A simple example of probability theory is the flipping of a coin. The probability of flipping a coin and it landing heads up is 0.5 (or 50%). This means that if a coin is flipped 100 times, it would be expected to land heads up around 50 times. This is an example of the Law of Large Numbers in action. Additionally, if two coins were flipped simultaneously, the probability of both coins landing heads up would be 0.25 (or 25%), which is calculated by multiplying the probability of each individual coin (0.5 x 0.5). This is an example of the Multiplication Rule.

## Formula of Probability theory

The formula for calculating the probability of an event occurring is given by\[P(A)=\frac{n(A)}{n(S)} \]

Where P(A) is the probability of the event occurring, n(A) is the number of favourable outcomes, and n(S) is the total number of possible outcomes.

## When to use Probability theory

Probability theory is useful in a variety of situations where there is uncertainty. It can be used to analyze the likelihood of certain events occurring and can help to inform decisions. Probability theory is used in a wide range of areas, including finance, economics, business, and engineering. It is also used in areas such as medical diagnosis, weather forecasting, and quality control. In particular, probability theory can be used to determine the probability of an event occurring given certain conditions, or to determine the probability of multiple events occurring simultaneously.

## Types of Probability theory

Types of probability theory include classical, frequentist, subjective and Bayesian probability.

**Classical Probability Theory**: This type of probability is based on equally likely outcomes, such as the probability of flipping a coin and getting heads or tails.**Frequentist Probability Theory**: This type of probability is based on the frequency of an event occurring, such as the probability of rolling a 6 on a die.**Subjective Probability Theory**: This type of probability is based on personal beliefs and opinions, such as the probability of winning a lottery.**Bayesian Probability Theory**: This type of probability is based on Bayes' Theorem, which states that the conditional probability of an event occurring is equal to the product of the probability of the event and the probability of the event occurring given the other events.

## Advantages of Probability theory

- Probability theory allows us to quantify the uncertainty associated with an event, which can be useful for making decisions.
- It can help us to understand how two events are related and the probability of them happening together.
- Probability theory can be used to make predictions about the outcomes of experiments and to test hypotheses about the underlying causes of events.
- It can be used to calculate the expected values of random variables and to calculate the expected return from a given investment.

## Limitations of Probability theory

Probability theory is a powerful tool that can be used to make decisions and understand uncertainty, but it also has several limitations. These limitations include:

- The assumptions of probability theory are not always accurate. For instance, the Law of Large Numbers assumes that each trial of an experiment is independent of the other trials, which may not always be true.
- Probability theory is limited by the information available. If the data used to calculate probabilities is incomplete or incorrect, the results of the analysis may be inaccurate.
- Probability theory is limited by its ability to predict the future. Since probability theory is based on past data, it cannot account for changes in the future that may affect the outcome of an experiment.

**Markov Chains**: This is a type of mathematical system that models transitions from one state to another based on certain probabilities. It is used to analyze the behavior of systems that evolve over time and can be used to predict the future state of the system.**Stochastic Processes**: This is a type of mathematical process that involves random variables. It can be used to model the uncertain behavior of systems over time and is often used to simulate the behavior of real-world systems.**Monte Carlo Method**: This is a type of numerical simulation technique which uses random numbers to solve problems. It can be used to calculate the expected value of a random variable and can also be used to estimate probabilities and expected outcomes.

In summary, other approaches related to probability theory include Markov Chains, Stochastic Processes, and the Monte Carlo Method. Markov Chains are used to model transitions from one state to another based on certain probabilities, while Stochastic Processes involve random variables to model the uncertain behavior of systems over time. The Monte Carlo Method is a numerical simulation technique which uses random numbers to solve problems and estimate probabilities and expected outcomes.

## Suggested literature

- Daniel J.,(2012),
*Sampling Essentials: Practical Guidelines for Making Sampling Choices*, Sage Publications ltd, p.65-79 - Hájek J., (1959),
*Optimal strategy and other problems in probability sampling*, Časopis pro pěstování matematiky Vol. 84 No. 4, p. 387-423 - Heavey E., (2011),
*Statistics for Nursing*, Jones & Bartllet Learning, p.65-66 - Humenik J., Hayne D., Overcash M., Gilliam J., Witherspoon A., Galler W., Howells D., (1980),
*Probability Sampling to Measure Pollution from Rural Land Runoff*, North Carolina State University Raleigh, p. 41-45 - Tansey O., (2007),
*Process Tracing and Elite Interviewing: A Case for Non-probability Sampling*, Political Science and Politics Volume 40 - Thomas R.,(1985),
*Estimating Total Suspended Sediment Yield With Probability Sampling*, Water Resources Research vol.21 no. 9, p.1381-1384