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Exploring Probability: Definitions, Properties, and Applications

Ayofemi Ovie^{* *}Correspondence: Ayofemi Ovie, Department of Mathematics, University of Calabar, Cross River State, Nigeria, Email:

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Description

Probability is a branch of mathematics that deals with the study of random events and the likelihood of their occurrence. It is a key concept in statistics and has numerous applications in various fields, including science, engineering, finance, and social sciences. Probability theory provides a formal framework for understanding uncertain events and making decisions based on them. In this article, we will discuss some of the key definitions and properties of probability theory.

Probability definition

Probability is defined as the likelihood or chance of an event occurring. The probability of an event is a number between 0 and 1, where 0 indicates that the event will not occur, and 1 indicates that the event will definitely occur. For example, the probability of flipping a coin and getting heads is 0.5 since there are two equally likely outcomes (heads or tails) and only one of them is heads.

Event definition

An event is a set of possible outcomes of an experiment. For example, if we toss a coin, the possible outcomes are heads or tails. The event of getting heads is a subset of the sample space (the set of all possible outcomes), and we can assign a probability to this event.

Sample space definition

The sample space is the set of all possible outcomes of an experiment. For example, if we toss a coin, the sample space is (heads, tails). The sample space can be finite or infinite, depending on the experiment.

Union definition

The union of two events A and B is the event that occurs if either A or B occurs (or both). We denote the union of A and B by A ∪ B. For example, if A is the event of getting heads when tossing a coin, and B is the event of getting tails, then A ∪ B is the event of getting either heads or tails.

Intersection definition

The intersection of two events A and B is the event that occurs if both A and B occur. We denote the intersection of A and B by A ∩ B. For example, if A is the event of getting heads when tossing a coin, and B is the event of getting heads again, and then A ∩ B is the event of getting heads twice in a row.

Complement definition

The complement of an event A is the event that occurs if A does not occur. We denote the complement of A by A'. For example, if A is the event of getting heads when tossing a coin, then A' is the event of getting tails.

Conditional probability definition

Conditional probability is the probability of an event A occurring given that another event B has occurred. We denote the conditional probability of A given B by P(A|B). For example, if we know that a card drawn from a deck is a heart, the probability of drawing an ace given that the card is a heart is the conditional probability P(ace|heart).

Independence definition

Two events A and B are independent if the occurrence of one does not affect the probability of the other occurring. In other words, P(A|B)=P(A) and P(B|A)=P(B). For example, if we toss a coin twice, the event of getting heads on the first toss is independent of the event of getting tails on the second toss.

Bayes' theorem definition

Bayes' theorem is a formula that describes the probability of an event A given evidence B. It is given by:

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

Where;

P(A)=Prior probability of A.
P(B|A)=Likelihood of B given A.
P(B)=Marginal probability of B.
P(A|B)=Posterior probability of A given B.

Bayes' theorem is widely used in Bayesian statistics and machine.