1.4.2 Infinitesimals in probability

So let’s go over the above discussion using calculus. If you have a probability density $P(x)$ for a continuous variable $x$, say the length of a blade of grass, or the weight of a person, or the energy of a particle, what’s probability you’ll find it with a value between $x$ and $x+dx$? Here $dx$ is an infinitesimal, in practice this means a very small number. This is the same as the problem you’re confronted with if you have a mass-density $\rho (x)$ that’s a function of position $x$. The answer is the same.

So now what’s the probability of lying between $x_1$ and $x_2$? That’s just like asking what’s the total mass between two points when the density as a function of position is $\rho (x)$. Integration was invented to solve that problem, by adding up all the infinitesimal $dx$’s.

Often this is used in statistics when determining how likely a result is to be incorrect. We’ll return to this point later.

One important point to realize is that the total area under the curve which from above, is the same as

That says the probability that $x$ has some value is $1$. That’s like saying the probability that you’re some place in the universe is $1$. If you ever start having doubts about this, be careful who you tell.