Abstract: A sequence of random variables is said to be exchangeable if permuting the indices of these random variables leaves their joint distribution unchanged. For example, a well-shuffled deck of cards is exchangeable because one extra shuffle doesn't change the (uniform) distribution of the order in which the cards will appear when dealt. More generally, a random object is exchangeable if its distribution is invariant for the natural action of the symmetric group (whatever this may be -- we'll give a few examples in the talk). De Finetti's theorem characterizes every infinite exchangeable sequence of random variables as a mixture of Bernoulli coin-tossing processes. I'll give an introduction to this circle of ideas and provide some extensions of this theorem characterizing complex random objects such as exchangeable set partitions, compositions of integers, and certain random trees derived from total partitions.
