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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?

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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 $1-\exp(-P\Delta t)\approx P\Delta t$.

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