Conditional probability (Bayes' Theorem)

Bayes’ theorem provides the algebraic formula for computing P(A|B)—the probability of event A given evidence B—directly from known probabilities.

Bob possesses three coins: two fair, one biased (heads 2/3, tails 1/3). He randomly selects one, flips heads—what’s the probability he chose the biased coin?

No matter how many consecutive heads occur, the probability the coin is unfair approaches but never reaches 100%, preserving fundamental uncertainty in inductive inference.

A person possesses two coins—one fair, one double-sided (both heads)—randomly selects one, flips it, and reports “heads.” What probability did they choose the fair coin?

Conditional probability problems become tractable through probability tree methodology—systematically growing branches representing sequential events and their possible outcomes.

When the coin-flipper reports a second consecutive “heads,” the probability tree extends further, and confidence in the fair coin decreases from 1/3 to 1/5.

When outcome probabilities differ across branches (fair coin 1/2-1/2, biased coin 2/3-1/3), tree balancing using least common multiples enables simple counting of equally likely leaves.

When new evidence emerges, probability trees require “trimming”—cutting branches leading to outcomes contradicted by observations.