More Than One Way to Be Fair Probability

An event is simply a subset of the sample space -- and this lets probability borrow set theory's entire vocabulary directly: the complement of A ('not A') is everything in the sample space that isn't in A, the union of A and B ('A or B') combines both, the intersection ('A and B') captures only what's shared between them, and mutually exclusive events are ones that can never both happen at once, meaning their intersection is empty.

Rather than starting from 'every outcome is equally likely,' the modern, axiomatic definition of probability requires only three things: every event's probability must be non-negative, the probability of the whole sample space must equal exactly 1, and the probability of any union of mutually exclusive events must equal the sum of their individual probabilities. Genuinely different probability assignments -- even ones that don't treat every outcome as equally likely, like a deliberately unfair coin -- can still satisfy all three rules and count as perfectly valid, showing that 'equally likely' is just ONE special case among many valid probability assignments, not a requirement baked into the definition itself.

Testing several candidate probability tables against the three axioms reveals which are genuinely valid: a table whose probabilities sum to 2.1 fails immediately (violating the sample-space-equals-1 rule), and a table containing even one negative value fails just as immediately (violating the non-negativity rule) -- while a table that passes all three checks, however strangely it distributes probability among outcomes, counts as a fully legitimate probability assignment.

Finding the probability of 'A or B' by simply adding P(A) and P(B) directly double-counts whatever overlaps between them -- the correct addition rule subtracts that overlap back out: P(A or B) equals P(A) plus P(B) minus P(A and B). A student exam-qualification problem shows this rule combined with the algebra of events (specifically De Morgan's law) in action: finding the probability that NEITHER of two students qualifies, that AT LEAST one fails, or that EXACTLY one qualifies, all follow directly from careful use of the complement, union, and intersection rules together.

Probability's origins trace to a 1654 dice question a gambler brought to Blaise Pascal, who worked through it in correspondence with Pierre de Fermat -- a genuinely famous exchange usually credited as the real starting point of probability theory, even though Gerolamo Cardano had already written a book on the subject roughly a century earlier, published only after his death. Centuries of further work, through Jacob Bernoulli, Abraham de Moivre, and Pierre-Simon Laplace, eventually led to Andrei Kolmogorov's 1933 axiomatic foundation -- the exact three-rule definition this chapter itself teaches, formalising a genuinely old set of gambling questions into rigorous modern mathematics.