To my mind one of the biggest open problems in probability, in the sense of being a famous basic statement that we don't know how to solve, is to show that there is "no percolation at the critical point" (mentioned in particular in section 4.1 of Gordon Slade's contribution to the Princeton Companion to Mathematics). A capsule summary: write $\mathbb{Z}_{d,p}$ for the random subgraph of the nearest-neighbour $d$-dimensional integer lattice, obtained by independently keeping each edge with probability $p$. Then it is known that there exists a critical probability $p_c(d)$ (the percolation threshold}) such that for $p < p_c$, with probability one
$\mathbb{Z}_{d,p}$ contains no infinite component, and for $p > p_c$, with probability one there exists an unique infinite component.
The conjecture is that with probability one, $\mathbb{Z}_{d,p_c(d)}$ contains no infinite component. The conjecture is known to be true when $d =2$ or $d \geq 19$.
Incidentally, one of the most effective ways we have of understanding percolation -- a technique known as the lace expansion, largely developed by Takeshi Hara and Gordon Slade -- is also one of the key tools for studying self-avoiding walks and a host of other random lattice models.
That article of Slade's is in fact full of intriguing conjectures in the area of critical phenomena, but the conjecture I just mentioned is probably the most famous of the lot.
Two which are for food rather than cash:
Let $f = t^{2d} + f_1 t^{2d-1} + f_2 t^{2d-2}+ \cdots f_d t^d + \cdots+ f_2 t^2 +f_1 t + 1$ be a palindromic polynomial, so the roots of $f$ are of the form $\lambda_1$, $\lambda_2$, ..., $\lambda_d$, $\lambda_1^{-1}$, $\lambda_2^{-1}$, ..., $\lambda_d^{-1}$. Set $r_k = \prod_{j=1}^d (\lambda_j^k-1)(\lambda_j^{-k} -1)$.
Conjecture: The coefficients of $f$ are uniquely determined by the values of $r_1$, $r_2$, ... $r_{d+1}$.
Motivation: When computing the zeta function of a genus $d$ curve over $\mathbb{F}_q$, the numerator is essentially of the form $f$. (More precisely, it is of the form $q^d f(t/\sqrt{q})$ for $f$ of this form.) Certain algorithms proceed by computing the $r_k$ and recovering the coefficients of $f$ from them. Note that you have to recover $d$ numbers, so you need at least $r_1$ through $r_d$; it is known that you need at least one more and the conjecture is that exactly one more is enough.
Reward: Sturmfels and Zworski will buy you dinner at Chez Panisse if you solve it.
Consider the following probabilistic model: We choose an infinite string, call it $\mathcal{A}$, of $A$'s, $C$'s, $G$'s and $T$'s. Each letter of the string is chosen independently at random, with probabilities $p_A$, $p_C$, $p_G$ and $p_T$.
Next, we copy the string $\mathcal{A}$ to form a new string $\mathcal{D}_1$. In the copying process, for each pair $(X, Y)$ of symbols in $\{ A, C, G, T \}$, there is some probability $p_1(X \to Y)$ that we will miscopy an $X$ as a $Y$. (The $16$ probabilities stay constant for the entire copying procedure.)
We repeat the procedure to form two more strings $\mathcal{D}_2$ and $\mathcal{D}_3$, using new probability matrices $p_2(X \to Y)$ and $p_3(X \to Y)$.
We then forget the ancestral string $\mathcal{A}$ and measure the $64$ frequencies with which the various possible joint distributions of $\{ A, C, G, T \}$ occur in the descendant strings $(\mathcal{D}_1, \mathcal{D}_2, \mathcal{D}_3)$.
Our procedure depended on $4+3 \times 16$ inputs: the $(p_A, p_C, p_G, p_T)$ and the $p_i(X \to Y)$. When you remember that probabilities should add up to $1$, there are actually only $39$ independent parameters here, and we are getting $63$ measurements (one less than $64$ because probabilities add up to $1$). So the set of possible outputs is a semialgeraic set of codimension $24$.
Conjecture: Elizabeth Allman has a conjectured list of generators for the Zariski closure of the set of possible measurements.
Motivation: Obviously, this is a model of evolution, and one which (some) biologists actually use. Allman and Rhodes have shown that, if you know generators for the ideal for this particular case, then they can tell you generators for every possible evolutionary history. (More descendants, known sequence of branching, etc.) There are techniques in statistics where knowing this Zariski closure would be helpful progress.
Reward: Elizabeth Allman will personally catch, clean, smoke and ship an Alaskan Salmon to you if you find the generators. (Or serve it to you fresh, if you visit her in Alaska.)
Best Answer
These are some big problems I know about:
$e$-positivity of Stanley's chromatic-symmetric functions for incomparability graphs obtained from $3+1$-avoiding posets. Shareshian and Wachs have some recent results related to this that connects these polynomials to representation theory, and they refine this conjecture with a $q$-parameter. Note that Schur-positivity is already known, and that the $e_\lambda$ expands positively into Schurs. It all boils down to finding a partition-valued combinatorial statistic on certain acyclic orientations.
Give a combinatorial description of the Kronecker coefficients.
Find a combinatorial description of the multiplicative structure constants for the Schubert polynomials (analogue of the Littlewood-Richardson coefficients in the Schur polynomial case).
The different variants of the shuffle conjecture. UPDATE: There is a recent proof on arxiv. There are still some generalizations of this that remains unproved.
Prove that LLT polynomials have positive Schur expansion. This is related to the shuffle conjecture and the $qt$-Kostka polynomials, see N. Loehr's notes.
Give a combinatorial formula for the non-homogeneous symmetric Jack polynomials (similar to the Knop-Sahi formula for the ordinary Jack polynomials).
Give combinatorial descriptions of structure constants that appear in plethystic substitutions, $a^\nu_{\lambda\mu} = \langle s_\lambda[s_\mu], s_\nu \rangle$
Find a combinatorial description of the $qt$-Kostka polynomials.
Find a combinatorial description of the polynomials $c^\nu_{\lambda\mu}(t)$ in $$s_\lambda(t) \cdot P_\mu(x;t) = \sum_\nu c^\nu_{\lambda\mu}(t) P_\mu(x;t)$$ where $P_\lambda(x;t)$ is the Hall-Littlewood polynomial.
Prove (or disprive) that the map $k \mapsto K_{k\lambda,k\mu}$ is a polynomial with non-negative coefficients, where $K_{\lambda\mu}$ is the Kostka coefficient. Or in more general, same question for $k \mapsto c^{k\nu}_{k\lambda,k\mu}$ for the Littlewood-Richardson coefficients. This question goes back to King, Tollu and Toumazet. Note that a proof of the latter would imply the famous saturation conjecture proved by A. Knutson and T. Tao. The conjecture is stated in Rassart's paper, here, where the polynomiality property is proved.
I have looked on the special case concerning Koskta coefficients, and this lead me to ask this somewhat related question. The negative answer to this question hints that the positivity conjecture might possibly be false, but a minimal counter example lives in a very high dimension, out of reach by exhaustive computer search. This concerns Ehrhart polynomials for non-integral polytopes, and very little is know about this case. One can show that such Ehrhart (quasi)polynomials have properties that Ehrhart polynomials corresponding to integral polytopes lack.
Whenever one seeks a combinatorial description, it is understood that the polynomial or number in question is conjectured to be a polynomial with non-negative integer coefficients.