In quantum mechanics, the expectation value of a observable $A$ is defined as $$\int\Psi^*\hat A\Psi$$
But in probability theory the expectation is a property of a random variable, with respect to a probability distribution:$$E(X):=\int X\;d\mu$$
I can't see how probability theory can be adapted to quantum mechanics. Observables are associated with linear operators, not measureable functions, so how can we talk about the expectation of a linear operator? And quantum mechanics textbooks use expectations and variances without mentioning underlying probability spaces. Does quantum mechanics use something other than ordinary probability theory?