exchangeable random variables
Sometimes a sequence of observations are not independent, yet they are perfectly symmetric — their labels (first, second, third) carry no special meaning, and any reordering would look statistically identical. Exchangeability captures exactly this 'order does not matter' kind of symmetry, which is weaker than independence but far more common in real modelling.
Formally, X_1, ..., X_n are exchangeable if their joint distribution is unchanged under every permutation of the indices: the probability of any pattern of values is the same no matter which slots you assign them to. Independent and identically distributed variables are always exchangeable (reordering independent identical things changes nothing), but the reverse fails: exchangeable variables can be strongly dependent. The defining feature is symmetry among the variables, not absence of dependence — they are identically distributed and share the same correlation between every pair.
Exchangeability is the natural assumption when units are interchangeable but possibly linked through a shared unknown — the classic example being draws whose common success rate is itself uncertain. De Finetti's theorem makes this precise and beautiful: an infinite exchangeable binary sequence behaves exactly as if you first drew a hidden probability p from some distribution and then made independent coin flips with that p. This is the bridge that turns subjective Bayesian priors into a consequence of symmetry, and it underlies hierarchical models throughout statistics.
Draw 3 balls without replacement from an urn with 2 red and 2 white. The colours X_1, X_2, X_3 are NOT independent (each draw changes the urn), yet they are exchangeable: P(red, white, red) = P(red, red, white) = P(white, red, red), because only the multiset of colours, not the order, affects the probability.
Without-replacement draws are exchangeable but dependent — order does not matter, yet the draws are linked.
Exchangeable is weaker than independent: i.i.d. implies exchangeable, but exchangeable variables are usually dependent. Exchangeability is a symmetry, not a lack of dependence.