Show there exists a turing machine with the following properties I'm struggling to understand a question I've been given. The question asks:
Let $\psi$ be a boolean formula in $n$ variables. There are $2^n$ different combinations of assigning
values to the variables. Consider the problem of deciding whether (strictly) more than $2^{n−1}$
of these assignments satisfy the formula $\psi$. We will call the language that corresponds to this
decision problem, $L$.

From this I can tell that if $x_1, x_2, ..., x_n$ are the $n$ variables which can be either true or false. I understand why there would be $2^n$ different assignings for the formula, as each variable can be assigned 1 of 2 values. But then what is $\psi$ exactly, is it an assignment such as $\psi$=($x_1 \lor x_2 \lor ... x_n$). But then the later part of the question doesn't make any sense. Can someone please explain in more detail what it means by "deciding whether (strictly) more than $2^{n−1}$ of these assignments satisfy the formula $\psi$"?

The question is then to show that there exists a turing machine $M$ and polynomials $T$ and $p$, with the following properties: 

• For every input $x$, $M$ terminates after at most $T(|x|)$ steps.

• If $x\in L$, then $Pr_{t∈\small\{0,1\small\}^{p(|x|)}}$[$M$ accepts $&lt;x, t&gt;$] $&gt;$ 1/2.

• If $x\notin L$, then $Pr_{t∈\small\{0,1\small\}^{p(|x|)}}$[$M$ rejects $&lt;x, t&gt;$] $≥$ 1/2

Where $Pr_{t∈\{0,1\}^{p(|x|)}}$[$M$ accepts $&lt;x, t&gt;$] means the probability, for a give $t$ from the set $\{0,1\}^{P(|x|)}$, that M accepts the input $&lt;x,t&gt;$ is greater than 1/2. 
This is similar to the definition of bounded error probabilistic polynomial time(BPP), except that the definition for BPP have both equalities as $≥$, so I'm 
guessing I need to show that the language L is in BPP. But how would I even start the proof to show that he language is indeed in BPP. Also the definition of a language in BPP is not identical to what is mentioned in the question, so maybe there's a different approach to answering the question. Also, I don't need to explicitly find the polynomials $p$ and $T$, but instead argue that they exist.

Any help to assist me with the question would be much appreciated
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.