Section 4.1

Relative Order of Exchangeable Values

Every ordering of i.i.d. continuous values, or of shuffled cards, is equally likely: increasing order, which of several items appears first, conditional order statements, ties versus strict wins between two dice.

Toolkit

Exchangeability is the key idea: if $X_1,\dots,X_n$ are i.i.d. (or sampled without replacement, or uniformly shuffled), then every one of the $n!$ relative orderings of their values is equally likely, regardless of the underlying distribution. Continuity removes ties, so each ordering has probability $1/n!$ ; with a discrete law, ties break this symmetry and must be handled separately (strict order has smaller probability than $1/n!$ suggests). Useful consequences:

$\mathbb{P}(X_1<X_2<\cdots<X_n)=1/n!$ for exchangeable continuous $X_i$ .
$\mathbb{P}(\text{a given one of }n\text{ items is the minimum/maximum}) = 1/n$ .
Only the relative order (ranks), not the actual values, determines events like “first occurrence,” “which appears before which,” or “is inside another.”
Conditioning on a sub-event about order reduces to counting permutations consistent with it, divided by total permutations.
For two independent discrete variables, $\mathbb{P}(X>Y)=\mathbb{P}(X<Y)=\tfrac12(1-\mathbb{P}(X=Y))$ .

Problems