Why do we use probability density, instead of probability?

Question

In https://en.wikipedia.org/wiki/It%C3%B4's_lemma

Under the section of Poisson jump processes, it is said that

We may also define functions on discontinuous stochastic processes. Let h be the jump intensity. The Poisson process model for jumps is that the probability of one jump in the interval [t, t + Δt] is hΔt plus higher order terms.

How can I know what the higher order terms are if we are just given the probability density being $h$? Is jump intensity (probability density) not accurate/sufficient?

Let's say the probability inside $(Δt)$ is

$h(Δt)+h(Δt)^3$.

Therefore the probability density is

$h$, as we keep to first order only.

Therefore probability density hides some information from probability, so why do we still use probability density as it is inaccurate?

Answer 1

We use it because $P(X=x)=0$ for all $x$ when $X$ is a continuous random variable. Because of that we need to define probability densities, i.e. "probabilities per foot" and they depend on units of $X$. They do not "hide" anything, but rather they are the only way we can access the information about relative probability densities for different $x$'s. Please refer to the great linked threads to learn more.