Let `alpha dt` be the probability of a sgas molecule experiencing a collision during the time interval `dt , alpha` is a constant. Find :
(a) the probability of a molecule experiencing no collisions during the time interval `t` ,
(b) the mean time interval between successive collisions.

Explanation

(a) Let `P(t)` = probability of no collision in the interval `(0, t)`, Then
`P(t + dt) = P (t) (1 - alpha dt)`
or `(dP)/(dT) = - alpha (t)` or `P (t) = e^(- alpha t)`
where we have used `P(0) = 1`
(b) The mean interval between collision is also the mean interval of no collision. Then
`lt t gt = (int_0^oo te^(- alpha t) dt)/(int_0^oo e^(-alpha t)) = (1)/(alpha) (Gamma (2))/(Gamma(1)) = (1)/(alpha)`.