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.
Addition Law of Probability
 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.