Knowing the decay constant $\lambda$ of a nucleus, find 
(a) the probability of decay of the nucleus during the time from $0$ to $t$;
(b) the mean lifetime $\tau$ of the nucleus.

(a) The probability of survival (i.e., not decaying) in time $t$ is $e^{-\lambda t}$. Hence the probability of decay is $1-e^{-\lambda t}$.

(b) The probability that the particle decays in time $dt$ around time $t$ is the difference 
    $e^{-\lambda t}-e^{-\lambda (t+dt)}=e^{-\lambda t}[1-e^{-e\lambda dt}]=\lambda e^{-\lambda t}dt$ 

Therefore the mean life time is 

     $T={\int }_0^{\infty }t\lambda e^{-\lambda t}dt/{\int }_0^{\infty }\lambda e^{-\lambda t}dt=\frac{1}{\lambda }{\int }_0^{\infty }x{e}^{-x}dx/{\int }_0^{\infty }{e}^{-x}dx=\frac{1}{\lambda }$