Probability at specific point in normal distribution curve

Question

In a normal distribution probability curve, it is said that the probability exists for a range of random variables. Probability for a specific point is said to be zero. Can you explain why is this so along with its practical/physical meaning?

Answer 1

There are a number of ways to look at this.

First, intuitively, the "size" of a single point in comparison to the real line is negligible. As an analogy, consider a population of size $10^{500}$. If we choose one element uniformly at random, then that single element occurs with probability $10^{-500} \approx 0$. The real line is "infinitely bigger" than a population of size $10^{500}$.

Formally, this is a consequence of the integral definition. For a continuous random variable $X$ with probability density function $f$, we define $$P(a \leq X \leq b) = \int_a^b f(x)\ dx.$$ If we only care about a single point, then we get $$P(X = a) = P(a \leq X \leq a) = \int_a^a f(x)\ dx = 0.$$ This holds for any continuous random variable, and normally distributed ones in particular.

If you want to get really technical, then this is a consequence of the fact that finite sets have Lebesgue measure zero. In a sense, sets like $\{a\}$ are simply too small to integrate over without getting $0$.