## Introduction

Probability is a branch of mathematics that deals with the study of random events or occurrences. It is widely used in various fields such as statistics, finance, and engineering. Probability theory is based on the study of the likelihood of events occurring and the calculation of the expected outcomes of those events.

## Mathematical Definition of Probability

In mathematics, probability is defined as a measure of the likelihood of a random event or occurrence. It is represented by a number between 0 and 1, with 0 indicating that the event is impossible and 1 indicating that the event is certain.

If we consider an experiment or random event E, the probability of E is denoted as P(E) and is defined as:

**P(E) = N(E) / N(S)**

where N(E) is the number of outcomes in event E and N(S) is the total number of possible outcomes in the sample space.

The probability of an event is the ratio of the number of outcomes in the event to the total number of outcomes in the sample space.

**For example,** consider rolling a fair six-sided die. The sample space is {1,2,3,4,5,6}, and the probability of rolling a 3 is 1/6, since there is one outcome that corresponds to rolling a 3 and there are six possible outcomes in the sample space.

## Probability of an Event

In probability theory, the probability of an event is a measure of the likelihood that the event will occur. It is represented as a number between 0 and 1, with 0 indicating that the event is impossible and 1 indicating that the event is certain to occur.

The probability of an event is usually denoted by P(E), where E is the event. It is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes:

**P(E) = Number of favorable outcomes / Total number of possible outcomes**

**For example,** if you roll a fair six-sided die, the probability of rolling a 2 is 1/6, since there is one favorable outcome (rolling a 2) and six possible outcomes (rolling any number from 1 to 6).

The probability of an event can also be expressed as a percentage or decimal. **For example,** a probability of 0.5 is equivalent to a 50% chance of the event occurring.

### Complementary Events

In probability theory, complementary events are two events that together encompass all possible outcomes of an experiment or situation. The complement of an event A, denoted by A’, is the event that consists of all outcomes that are not in A.

Mathematically, the complement of an event A is defined as:

A’ = {x | x is an outcome in the sample space S, but not in A}

The probability of the complement of an event A is denoted by P(A’), and can be calculated as:

**P(A’) = 1 – P(A)**

This means that the probability of either event A or its complement A’ must equal 1, since they are the only two possible outcomes. **For example,** if the probability of event A is 0.7, then the probability of its complement A’ is 1 – 0.7 = 0.3.

## Probability Terms and Definition

Probability Term | Definition |
---|---|

Sample Space | The set of all possible outcomes of an experiment or situation. |

Event | A subset of the sample space, representing a particular outcome or set of outcomes. |

Probability | A measure of the likelihood that an event will occur, represented as a number between 0 and 1. |

Complementary Events | Two events that together encompass all possible outcomes of an experiment, with the complement of one event being the set of outcomes not in that event. |

Conditional Probability | The probability of an event occurring given that another event has already occurred. |

Independent Events | Two events that do not affect each other’s probability of occurring. |

Dependent Events | Two events where the occurrence of one affects the probability of the other occurring. |

Mutually Exclusive Events | Two events that cannot occur at the same time. |

Union | The event that either one or both of two events occur. |

Intersection | The event that both of two events occur. |

Probability Distribution | A function that describes the probabilities of all possible outcomes in a sample space. |

Expected Value | The long-run average value of a random variable, weighted by its probability distribution. |

Variance | A measure of how spread out the values of a random variable are around its expected value. |

Standard Deviation | The square root of the variance, representing the average distance of each value from the expected value. |

## Applications of Probability

- Risk assessment and management
- Statistical inference
- Gaming and gambling
- Quality control
- Sports analytics
- Artificial intelligence and machine learning
- Epidemiology and public health

## Examples on Probability

**Tossing a coin:** When you toss a fair coin, the probability of getting heads or tails is 0.5 or 50% each.

**Rolling a die:** When you roll a fair six-sided die, the probability of getting any specific number (1, 2, 3, 4, 5, or 6) is 1/6 or approximately 16.67%.

## Frequently Asked Questions (FAQs) on Probability

**What is probability?**

Probability is a measure of the likelihood or chance of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates that the event is impossible and 1 indicates that the event is certain.

**How is probability calculated?**

The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For example, if you flip a coin, the probability of getting heads is 1/2 or 0.5, because there is one favorable outcome (getting heads) out of two possible outcomes (getting heads or tails).

**What is the difference between theoretical and experimental probability?**

Theoretical probability is the probability of an event based on mathematical calculations and assumptions. Experimental probability is the probability of an event based on actual observations or experiments.

**What is conditional probability?**

Conditional probability is the probability of an event occurring given that another event has occurred. It is calculated by dividing the probability of both events occurring by the probability of the given event occurring.

**What is the difference between independent and dependent events?**

Independent events are events in which the occurrence of one event does not affect the probability of the other event occurring. Dependent events are events in which the occurrence of one event affects the probability of the other event occurring.

**What are some real-world applications of probability?**

Probability has many real-world applications, including risk assessment, statistical inference, gaming and gambling, quality control, sports analytics, artificial intelligence and machine learning, and epidemiology and public health.