Show there exists a turing machine with the following properties
Look again at your definitions. $\psi$ is a formula of $n$ variables, eg.
$\psi_1 = x_1 \vee x_2$ or $\psi_2=x_1 \wedge x_2$, on 2 variables.

Now the question asks whether or not there are more than $2^{n-1}$ assignments to $x_1, x_2, ...$ that satisfy the formula. In other words, assignments that make $\psi$ evaluate to true.
For the above example, $\psi_1$ has 3 satisfying assignments (out of the $2^2=4$ possible assignments), while $\psi_2$ is satisfied only by a single assignment (namely, $x_1=x_2=\text{true}$). So $\psi_1 \in L$ and $\psi_2 \notin L$.

If you understand so far, then finding a probabilistic TM that accepts any formula with the above conditions with probability at least $1/2$ is trivial. I'll let you think about it some more.