Probability splits into two incompatible readings: the frequentist view that a probability is the long-run proportion of an event across many trials, and the Bayesian view that it is a measure of personal confidence given current knowledge. For one-off events that will never repeat, only the degree-of-belief reading can assign a number at all.
The 17th-century gambler Chevalier de Mere noticed he profited betting on at least one six in four rolls of a die but lost betting on at least one double-six in twenty-four rolls of two dice, though he had reasoned the two bets equivalent. His query to Pascal, who corresponded with Fermat about it, produced the first systematic calculation of compound event probabilities and helped found the formal theory.
A matching trait with a one-in-a-million random frequency is routinely presented as near-certain guilt, but this confuses the probability of the evidence given innocence with the probability of innocence given the evidence. In a large population many innocent people may also match, so the raw match frequency must be weighed against how many candidates the trait could implicate.
Genuinely random distributions produce apparent clumps, streaks, and coincidences, so patterns that look designed or ominous (disease clusters, lucky runs, bombing maps) are often what randomness predicts. Reading intention or cause into such clusters mistakes the ordinary texture of chance for signal.
To evaluate a gamble or decision, multiply each possible outcome's value by its probability and sum the results, yielding the long-run average return. This exposes why most casino and lottery bets are losing propositions even when a large payoff is possible, since the tiny win probability cannot offset the stake.