Probability & Counting Formulas

Basic Probability

P(A) = \frac{\text{favourable outcomes}}{\text{total outcomes}}

Basic Probability calculates the likelihood of an event by dividing favorable outcomes by the total number of possible outcomes.

Complement Rule

P(A^c) = 1 - P(A)

Find the probability of the opposite event by subtracting from 1 — use this when it's easier to calculate what doesn't happen.

Addition Rule

P(A \cup B) = P(A) + P(B) - P(A \cap B)

The Addition Rule calculates the probability that at least one of two events occurs by adding individual probabilities and subtracting their overlap.

Multiplication Rule

P(A \cap B) = P(A)\,P(B \mid A)

Find the probability that two events both occur when one might affect the chance of the other using multiplication of probabilities.

Independent Events

P(A \cap B) = P(A)P(B)

When events are independent, multiply their individual probabilities together to find the probability that both events occur.

Conditional Probability

P(A \mid B) = \frac{P(A \cap B)}{P(B)}

Conditional probability calculates the likelihood of one event given that another event has definitely already occurred, denoted P(A|B).

Bayes' Theorem

P(A \mid B) = \frac{P(B \mid A)P(A)}{P(B)}

Bayes' Theorem updates the probability of a hypothesis given observed evidence, answering what's true given what you now know.

Permutations

P(n, r) = \frac{n!}{(n-r)!}

Permutations count the number of ways to arrange a set of objects in a specific order, where changing the order creates a different arrangement.

Combinations

\binom{n}{r} = \frac{n!}{r!(n-r)!}

Combinations count the ways to select items from a group when order doesn't matter; they are used for choosing unordered subsets.

Fundamental Counting Principle

N = n_1 \times n_2 \times \cdots \times n_k

Multiply the number of choices at each step to find the total number of different possible outcomes when making sequential selections or decisions.

Permutations with Repetition

\frac{n!}{n_1!\,n_2!\cdots n_k!}

Counts the distinct arrangements of n total objects when some are identical; use this when items repeat in groups and order matters.

Binomial Probability

P(X = k) = \binom{n}{k}p^{k}(1-p)^{\,n-k}

Binomial Probability finds the likelihood of exactly k successes in n independent identical trials, each with fixed probability p.

Expected Value

E(X) = \sum x_i\,P(x_i)

Expected value is the average outcome you'd get if you repeated an experiment many times, weighted by how likely each outcome is.

Variance of a Random Variable

\operatorname{Var}(X) = E(X^2) - \big[E(X)\big]^2

Variance measures how spread out a random variable's values are around its mean, using the formula Var(X) = E(X²) - [E(X)]².

Geometric Probability

P(X = k) = (1-p)^{\,k-1}p

Geometric Probability tells you the probability that the first success in a series of independent trials occurs on exactly the k-th attempt.

Mutually Exclusive Events

P(A \cap B) = 0 \implies P(A \cup B) = P(A) + P(B)

When two events cannot happen at the same time, use this formula to find the probability that at least one of them occurs.

Odds

\text{odds} = \frac{P(A)}{1 - P(A)}

Odds tell you the ratio of the probability an event happens to the probability it doesn't, used when you want to compare likelihood of success to failure.

Law of Large Numbers

\bar{x}_n \to \mu \text{ as } n \to \infty

The Law of Large Numbers tells you that the sample mean gets arbitrarily close to the true population mean as your sample size increases.

Sample Space

S = \{ \text{all possible outcomes} \}

The sample space lists all possible outcomes of an experiment, forming the foundation for probability calculations and predictions.

Poisson Distribution

P(X = k) = \frac{\lambda^{k}e^{-\lambda}}{k!}

The Poisson Distribution gives the probability of a specific number of independent events occurring in a fixed interval when the average rate is known.

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