Bayesian vs. Frequentist Thinking

Frequentist probability defines probability as long-run frequency: the probability of heads is 0.5 because if you flip a coin infinitely many times, half will be heads. Bayesian probability defines probability as a degree of belief: the probability of rain tomorrow is 70% because, given current evidence, you believe there's a 70% chance. This distinction matters enormously in practice. A frequentist cannot assign a probability to a one-time event ('What is the probability this defendant is guilty?') because there is no long-run frequency—it's a unique case. A Bayesian can, using prior knowledge and evidence to quantify belief. Neither interpretation is 'wrong'; they answer different questions.