  Exchangeability is a fundamental concept in probability theory and
statistics. It allows to model situations where the order of observations does
not matter. The classical de Finetti's theorem provides a representation of
infinitely exchangeable sequences of random variables as mixtures of
independent and identically distributed variables. The quantum de Finetti
theorem extends this result to symmetric quantum states on tensor product
Hilbert spaces. However, both theorems do not hold for finitely exchangeable
sequences. The aim of this work is to investigate two lesser-known
representation theorems. Developed in classical probability theory, they extend
de Finetti's theorem to finitely exchangeable sequences by using
quasi-probabilities and quasi-expectations. With the aid of these theorems, we
illustrate how a de Finetti-like representation theorem for finitely
exchangeable sequences requires a mathematical representation which is formally
equivalent to quantum theory (with boson-symmetric density matrices).
