Probability & Combinatorics

Counting, permutations, combinations, and core probability laws.

Probability & Combinatorics brings together the counting tools and probability rules used to analyse chance, arrange outcomes, and make sense of uncertainty. From factorials, permutations, and combinations to conditional probability, Bayes’ theorem, and expected value, these formulas help students, exam takers, and professionals solve problems in statistics, data, science, finance, and everyday decision-making with confidence.

Permutations

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

Number of ordered arrangements of r items from n.

Combinations

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

Number of unordered selections of r items from n.

Factorial

n! = n \times (n-1) \times \cdots \times 2 \times 1

Product of all positive integers up to n.

Permutations with Repetition

n^r

Ordered arrangements of r items from n when repetition is allowed.

Permutations of a Multiset

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

Arrangements of n items where some are identical.

Classical Probability

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

Probability of an event with equally likely outcomes.

Complement Rule

P(A') = 1 - P(A)

Probability that an event does not occur.

Addition Rule

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

Probability that A or B occurs.

Multiplication Rule (Independent)

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

Probability of both independent events occurring.

Conditional Probability

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

Probability of A given that B has occurred.

Bayes Theorem

P(A|B) = \frac{P(B|A)\,P(A)}{P(B)}

Reverses conditional probabilities using prior knowledge.

Expected Value

E[X] = \sum_{i} x_i\, p_i

Long-run average value of a random variable.

Variance of a Random Variable

Var(X) = E[X^2] - (E[X])^2

Measures the spread of a random variable around its mean.

Binomial Probability

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

Probability of k successes in n independent trials.

Poisson Probability

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

Probability of k events in a fixed interval given average rate λ.

Geometric Probability

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

Probability that the first success occurs on trial k.

Normal Distribution PDF

f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}

Bell-curve probability density function.

Odds

Odds = \frac{P(A)}{1 - P(A)}

Ratio of the probability of an event to its non-occurrence.

Pigeonhole Principle

\lceil n/k \rceil

If n items go into k containers, some container holds at least this many.

Inclusion-Exclusion (3 sets)

|A\cup B\cup C| = |A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|

Counts the union of overlapping sets without double counting.

Catalan Number

C_n = \frac{1}{n+1}\binom{2n}{n}

Counts many recursive structures such as balanced parentheses.

Stars and Bars

\binom{n+k-1}{k-1}

Ways to distribute n identical items into k distinct groups.