I'm reading "Bayesian Data Analysis" by Gelman et al., and I encountered this exchangeability property: $\{X_n\}_{n \in N}$ is exchangeable if $F_{X_1,\ldots,X_n}(x_1,\ldots,x_n)$ is symmetric in its arguments $\forall n \in N$. I understand the definition, but not the intuition behind it. Up to now I've always only encountered i.i.d. sequences of random variables. I understand the intuition behind the i.i.d. property (for example, it's a reasonable model for coin tosses, dice throws, etc.) and its usefulness in forming various kinds of confidence intervals (mean, proportions, quantiles, regression coefficients, etc.).
I'm much more at a loss with exchangeability. Obviously i.i.d. sequences are exchangeable. But which other kind of phenomena are intuitively exchangeable, and how is this property used to perform inference? I read that an exchangeable sequence is one where the probability of a specific event (for example, with $p(X_1=1, X_2=0,\ldots,X_n=1)$ where the $X_i$ are Bernoulli) doesn't depend on the order of the results. But then sampling without replacement from a urn with $n$ black marbles and $m$ white marbles (which I read can be modeled by an exchangeable sequence of Bernoulli RVs) doesn't seem intuitively exchangeable to me, because I would think that the probabilities would depend on the results of the extractions. Probably it's the conditional probabilities which depend on the extraction history, and not the joint density, but I'm still confused...I would need some intuitive interpretation of exchangeability, and one or two simple examples where we use an exchangeable, but not i.i.d, sequence of random variables to perform statistical inference.