Counting in Probability

Definition

Counting in probability is the use of combinatorial counts to compute probabilities in finite sample spaces, especially when outcomes are equally likely.

Why It Matters

Many probability problems are hard only because the event space is described indirectly. Counting translates the problem into a ratio of favorable configurations to total configurations, which makes finite sampling, discrete distributions, randomized algorithms, and introductory Bayesian examples easier to reason about.

Formalism / Key Objects

• Finite equiprobable sample space:

$$P(A)=\frac{|A|}{|\Omega |}.$$

• Binomial coefficient:

$$\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{k!(n-k)!}.$$

• Binomial probability:

$$P(K=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}.$$

• Conditional counting:

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

when all outcomes inside B remain equally likely.

• See Finite Probability Counting Rules for the compact derivation of the binomial coefficient and Bernoulli-counting formulas.

Connections

• Core background for Probability and Statistics.
• Supports Floyd’s Sampling Algorithm, which samples exact k-subsets without enumerating all binom(n,k) possibilities.
• Connects to Beta Distribution through binomial likelihoods and probability models over Bernoulli rates.
• Finite Probability Counting Rules holds the reusable formula sheet and derivation.

Common Confusions

• Counting formulas only give probabilities directly when the counted outcomes are equally likely.
• n!, 2^n, and binom(n,k) count different objects: orderings, subsets, and fixed-size subsets.
• Conditional counting counts inside the conditioned event, not inside the original whole sample space.

Key Sources

• MIT6041 - Probability Counting
• Jaffray2026 - Floyd’s Sampling Algorithm