# Probability

## Fundamental Concepts of Probability

• Probability is a measure of the likelihood that a particular event will occur.
• Probabilities are expressed as numerical values ranging between 0 and 1, inclusive. A probability of 0 indicates that an event will certainly not occur, while a probability of 1 indicates that an event will certainly occur.
• An experiment is an action or process that leads to an outcome that is uncertain.
• Sample space (S) is the set of all possible outcomes of an experiment.
• An event is a subset of a sample space.

## Assigning Probabilities to Events

• The probability of an event E is calculated by the formula P(E) = n(E) / n(S), where n(E) is the number of outcomes favourable to E and n(S) is the total number of outcomes in the sample space.
• The law of total probability states that the sum of the probabilities of each possible outcome in the sample space is equal to 1. i.e., ΣP(E)=1.

• The addition law of probability is used when we want to find the odds of either of two events happening.
• It states that for any two events, A and B, the probability that either A or B will happen is the sum of the probability of each event, minus the probability of the two events overlapping, expressed as P(A U B) = P(A) + P(B) - P(A ∩ B).

## Multiplication Law of Probability

• The multiplication law of probability is used when we want to know the probability that two events will both occur.
• For independent events, it specifies that the probability of both events happening is the product of the probabilities of the two events, expressed as P(A ∩ B) = P(A) * P(B).

## Conditional Probability

•  Conditional Probability refers to the probability of one event occurring given that another event has already occurred. For events A and B, it’s given by the formula **P(A B) = P(A ∩ B) / P(B)**.

## Bayes Theorem

• Bayes’ theorem connects the conditional and marginal probabilities of two random events. It provides a way to update our initial beliefs in the light of new data, commonly known as performing a Bayesian update.

## Random Variables and Distributions

• A random variable is a variable whose value depends on outcomes of random phenomena. They can be discrete or continuous.
• The probability distribution of a random variable X is a function that assigns probabilities to different outcomes of X.
• The expected value (mean), variance and standard deviation are key properties of a distribution.