Section 4.3

Ingenious Symmetries and Couplings

Problems cracked by one clever symmetry or coupling: the airplane seat, Polya's urn, n+1 versus n coins, equal head counts, tournament brackets, strength-conserving duels.

Toolkit

Look for a symmetry or an exchangeable coupling that makes a seemingly order- or history-dependent question collapse to something trivial, rather than computing a sum or recursion directly.

Key devices:

Relabelling symmetry: if two outcomes are related by a measure-preserving relabelling (swap heads/tails, swap two players’ roles, reverse a sequence), they are equally likely; often reduces “A beats B” to $\tfrac12$ or to a count via $\binom{2n}{n}/4^n$ (Vandermonde).
Exchangeability / Pólya urn: sequences of successes and failures with the same total count are equiprobable regardless of order, so a running proportion can be uniform on its support.
Stopping-time coupling: track only the event that ultimately decides the outcome (e.g. which of two marked seats/items gets hit first); intermediate randomness is irrelevant once the two tracked events are symmetric.
Martingale conservation: if a quantity (strength, money) splits proportionally and each step is a fair re-randomisation, the win probability equals the initial share, independent of the sequence of steps.
Uniform random permutation: in knockout/elimination structures, the arrival or seeding order is a uniform permutation, letting you compute bracket events by counting permutations directly.