I have been following Feynman lectures on Physics, lecture 43 — Diffusion.
There he derives the average value of time till next collision over all molecules. He finds it to be equal to the average time between collisions $\tau$. This concept is pretty clear.
Then he goes on to derive the drift velocity of molecules (and in my case of interest, electrons). Here he states that the average time between collisions also to be equal to the average number of collisions per second. My doubt lies here:
I'm not able to mathematically prove how the average value of time since last collision to be equal to the average time between collisions.
I checked up some online resources too, none of them explain this properly.
How can it be proved mathematically?
Edit: Another question cropped up in this same topic. The author derives the probability that the molecule survives t seconds without collision is
Since all these events are independent shouldn't the integral of this probability function from $t=o$ to infinity should be equal to $1$? But if we evaluate this integral we will actually get $\tau$ as the answer. Isn't this a violation of the axioms of probability?