What is the probability that two data points in a normally distributed data set have the same value? How does one determine the probability that two data points have the same value, given the data set is normally distributed and how does the actual value of these points influence this probability?
 A: If the distribution is truly normal, then all the answers are real numbers. Let us take an example of a normal distribution, $X\sim\mathcal{N}(\mu,\sigma)$, the standard normal, $X\sim\mathcal{N}(0,1)$. Since bivariate normal distributions can be transformed to be a standard normal distribution, what we observe for the standard normal distribution applies to all other normal distributions as well in some alternative scale and location. This includes the difference between two normal distributions as one such linear transformation, i.e., $Z=X-Y$ is normal and where transformed $Z\rightarrow W$ and $W\sim\mathcal{N}(0,1)$. Thus the difference between two normal distributions reduces to the standard normal distribution, a well know fact used in one sample testing.
Suppose we wish to reproduce a value of an $X_i$ of $X\sim\mathcal{N}(0,1)$ that is as close to zero as possible, and as that is the mean value of the standard normal, and that is its maximum density value, i.e., having the most densely packed sample-data. We might obtain a value of 0.001, or -0.0005, but as we attempt to obtain a value that is exactly zero, all we can do is approach it more closely as we continue to acquire more and more random $X$-values. Now provided that we record the $X$-value as a real number with infinite precision, we will never reproduce a zero value exactly. So, the answer is, the probability is as small as we like, provided that the recorded precision is sufficiently large, which is a "nice" way of saying that the probability is zero.
Now, given that even when the density is at its maximum value, the probability is zero, it is certainly no larger a probability for other values, such that the answer is $p=0$ everywhere.
