Probability

Understanding Probability

  • Probability is a branch of mathematics that deals with predicting the likelihood of certain events occurring and modelling uncertain processes.
  • It calculates the chance of events to occur using a scale from 0 to 1 or from 0% to 100%, where 0% represents an impossible event and 100% represents a certain event.

Probability Concepts

  • An event is an outcome or combination of outcomes from a certain experiment or process.
  • The sample space (often denoted by S or Ω) is the set of all possible outcomes of the experiment.
  • The probability of an event A is usually denoted as P(A).

Classical Approach to Probability

  • In the classical (also known as theoretical) approach to probability, all outcomes are considered equally likely. The probability of an event is calculated as the ratio of the number of favourable outcomes to the total number of possible outcomes.

Empirical Approach to Probability

  • The empirical (or statistical) approach to probability is based on observations or experiments. The probability of an event is determined by dividing the number of times the event occurs by the total number of trials.

Mutually Exclusive and Independent Events

  • An event is considered mutually exclusive if the occurrence of one event excludes the occurrence of the other.
  • Events are independent if the occurrence of any one event does not affect the occurrence of the other.

Addition and Multiplication Rules

  • The addition rule is used to find the probability of the union of two events.
  • The multiplication rule is used to find the probability of the intersection of two events.

Conditional Probability

  • Conditional Probability is the probability of one event occurring given that another event has already occurred. It is denoted by P(A B), which reads as the probability of event A given event B.

Bayes’ Theorem

  • Bayes’ theorem is used to find the probability of an event given the prior knowledge of conditions that might be related to the event. It is an important concept in a wide range of fields including medicine, psychology, games, and law.

Applying Probability

  • Probability is used in numerous areas including statistics, finance, gambling, artificial intelligence, philosophy, machine learning, and computer science. Mastering probability concepts helps in developing logical thinking and problem-solving skills