Calculate the probability of mu for normal distribution Given we have drawn $n$ samples $x_1,...x_N$ from a normal distribution $\mathcal{N}(\mu, \sigma)$. How do I calculate the probability that eg $\mu=1$? ie $P(\mu=1|x_1,...x_N)$
Is it simply $1-\alpha$ of a T-distribution for $\mu=1$?
And taking this forward: How do I calculate the probability that this $\mu$ is drawn from a Normal distribution with e.g. $\mu_2=1$ and $\sigma_2=0.5$?
 A: The probability that $\mu=1$ given a sample with a non-degenerate prior density is zero.
$$\Pr(\mu|X,\sigma^2)=\int_0^\infty\frac{\prod_{i=1}^nf(X|\mu,\sigma^2)\pi(\mu,\sigma^2)}{\int_{-\infty}^\infty\int_0^\infty{\prod_{i=1}^nf(X|\mu,\sigma^2)\pi(\mu,\sigma^2)}\mathrm{d}\mu\mathrm{d}\sigma^2}\mathrm{d}\sigma^2,$$ where $f$ is the Gaussian likelihood function.
At $\mu=1$ the density function has height but zero width.
Alternatively, you could construct the same discussion by noting that a countable point in a continuum of countable points has zero measure and therefore zero probability.
To discuss the probability of $\mu$ having some value there has to be a range, even if very small.
The only exception to this would be a degenerate prior density or a prior with weight only on a countable set that caused the function to be countable.
My guess, though, is that this is not what you are really asking.  My guess is that you are really asking "what is the probability of seeing data as extreme or more extreme, given that $\mu=1$?"
That is $\Pr(X|\mu)$.  If the variance isn't known, then yes, that would be a Student t-distribution.  You would invert the t value from $$t=\frac{\bar{x}-1}{\sqrt{\frac{s^2}{n}}},$$ by looking it up on a table.  
If you were really discussing the Bayesian solution rather than the Frequentist or maximum likelihood solution, then you would be in the subjective interpretation of Bayesian methods where $\mu$ is drawn from the prior.  In that case $$\pi(\mu;\sigma^2)=\mathcal{N}(1,.25).$$ based on your edit of $\mu=1,\sigma=.5$.
EDIT
Now I think I know less of what you are asking than with the first question.  It may help if you describe the underlying data and what you need to infer.  If you were writing out a set of hypotheses in formal writing, what would they be? Skip the goal method, that is irrelevant.  The method should follow not lead.
A: The probability that $\mu=1$ is zero: $P(\mu=1|X)=0$ if you're drawing $\mu$ from a continuous distribution.
If you want to apply inference techniques, then it's usually for intervals, such as $\mu<1$ or $0<\mu<1$.
