The probability of an event \( A \) is calculated by dividing the number of favorable outcomes by the total number of possible outcomes: \[ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]

Independent events are those whose outcomes do not affect each other. For example, flipping a coin twice results in independent events. Dependent events are those where the outcome of one event affects the outcome of the other, such as drawing cards without replacement.

If two events, \( A \) and \( B \), are independent, the probability of both events happening is: \[ P(A \cap B) = P(A) \times P(B) \]

Conditional probability is the probability of an event occurring given that another event has already occurred. It is calculated as: \[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \] where \( P(A \mid B) \) is the probability of \( A \) given \( B \), \( P(A \cap B) \) is the probability of both \( A \) and \( B \), and \( P(B) \) is the probability of \( B \).

Bayes' Theorem is used to update the probability of a hypothesis based on new evidence. It is expressed as: \[ P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)} \] where \( P(A \mid B) \) is the posterior probability, \( P(B \mid A) \) is the likelihood, \( P(A) \) is the prior probability, and \( P(B) \) is the marginal likelihood.

The probability of at least one event happening is calculated using the complement rule: \[ P(\text{at least one}) = 1 - P(\text{none}) \] where \( P(\text{none}) \) is the probability that none of the events happen.

A probability distribution describes how the probabilities are distributed over the possible outcomes of a random variable. The most common distributions include the binomial distribution, normal distribution, and Poisson distribution.

The expected value (or mean) of a discrete random variable is calculated as: \[ E(X) = \sum X_i P(X_i) \] where \( X_i \) is each possible value of the random variable and \( P(X_i) \) is the probability of each value.

The law of large numbers states that as the number of trials increases, the sample mean will converge to the expected value. In other words, the larger the sample size, the closer the sample mean will be to the population mean.

Permutations refer to the arrangement of objects where the order matters. Combinations refer to the selection of objects where the order does not matter. The number of permutations of \( n \) objects taken \( r \) at a time is given by: \[ P(n, r) = \frac{n!}{(n - r)!} \] The number of combinations of \( n \) objects taken \( r \) at a time is given by: \[ C(n, r) = \frac{n!}{r!(n - r)!} \]