Given an observed sample from a binomial distribution and a known prior, how can I update the probability distribution of possible 'p' values? Let's say that I have an 'unfair' coin, for which I'm interested in estimating the 'heads' likelihood or 'p' value.
I've been told that the 'heads' likelihood for my 'unfair' coin is normally distributed around Z%.
How can I update the probability distribution for 'p' values after observing X heads out of Y trials?
 A: You seem to be asking about a problem in Bayesian inference;
You start with a prior on $p=P(\text{Head in a toss of the coin})$.
You have an experiment that will give a (presumably binomially distributed) number of heads, $X$, in $n$ tosses.
You want to update your prior with the result of the experiment (giving a posterior distribution which summarizes your information on $p$).
Note that, from Bayes' theorem, posterior $\propto$ likelihood $\times$ prior, or in this particular case:
$f(p|X=x)\propto f_X(x|p) f(p)$ 
where the likelihood is proportional to the conditional density of the variable given the parameter $f_X(x|p)$ (again, presumably this is a binomial, so trivial to evaluate). (Here I abuse notation a little, but hopefully it is clear)
Here's an illustration of a prior, a likelihood (normalized so that it fits on a similar scale) and posterior:

You can evaluate this $f(x|p) f(p)$ at any given value of $p$, and so scale to an actual posterior (by finding the normalizing constant, for example by numerical integration over $p$ between 0 and 1).
This could readily be done with a truncated normal prior if you wished.

For example, consider 
(i) a truncated normal prior on $\pi=P(H)$, which we'll base off a normal distribution with mean 0.6 and standard deviation 0.2 but then truncated to be between 0 and 1 (so the mean is a bit lower and the standard deviation a bit smaller). We can compare that with Bruce's suggestion of using a beta prior, here with mean 0.6 and standard deviation of 0.2. 
(ii) a sample of 32 tosses with 12 heads and 20 tails, which we model as binomial
Here are the priors and posteriors for that setup:

We see that the priors look a bit different (though broadly in the same place) while the posterior distributions look almost identical
par(mfrow=c(1,2))
nprior <- function(p) dnorm(p,.6,.2)  # set up the prior
bprior <- function(p) dbeta(p,3,2)
curve(nprior,0,1,main="Truncated normal prior (with beta)")
curve(bprior,0,1,col="darkgreen",lty=2,add=TRUE)

lik <- function(p) dbinom(12,32,p)

npost.un <- function(p) lik(p) * nprior(p)  # this is Bayes rule
k <- integrate(npost.un,0,1)$value
npost <- function(p) npost.un(p)/k  # normalize the posterior to a density
bpost <- function(p) dbeta(p,3+12,2+20)
curve(npost,0,1,main="Corresponding posteriors")
curve(bpost,0,1,col="darkgreen",lty=2,add=TRUE)

