In nuclear physics, the intensity of a particle source is measured with Geiger-Muller counters. A particle entering the counter generates a discharge in it that lasts time τ , during which the counter does not record any particles entering the counter. Find the probability that the counter will count all particles entering it during time t if the following conditions are fulfilled:
the particles enter the counter independently;
the probability that during the time interval from t to \(t + \Delta t .\) , k particles entered the counter is given by
\[ p _ { k } ( t , t + \Delta t ) = \frac { ( a \Delta t ) ^ { k } e ^ { - a \Delta t } } { k ! } ,\tag{43} \]
where a is the rate.