# Collision probability vs rate of collision

While studying the kinetic theory of matter I came across a term called collision rate ($P$) which is the number of collisions suffered by a molecule in unit time. Later it was stated that $P$ is also known as collision probability, as the probability that a molecule suffers collision in time $t$ is $Pt$. But isn't probabilty a fraction $\leq1$? The expression for collision probability should have been something like $P=\frac{\mathrm{collision\ rate}\cdot t}{\binom{n}{2}}$, where $n$ is the total number of molecules in the system. According to the book both the rate and probability are equal. Can you explain why? Or is it incorrect?

If we treat the collisions as a Poisson process then the probability of there being a collision within some interval $\Delta t$ is actually given by $1-\exp(-P\Delta t)$ which is less than or equal to one, as you'd expect.
What your book has then done is implicitly assume that $P\Delta t$ is small (certainly less than 1), giving rise to the approximation $1-\exp(-P\Delta t)\approx P\Delta t$.