A Geiger counter beeps according to a Poisson arrival process, with rate 1 beep per minute. Let T_3 be the time in minutes from when the Geiger counter is turned on until the third beep.
Find P( 1 < T_3 < 3 ) = probability T_3 falls in interval (1,3).
Can someone check which of my answers are correct?
Yes, if you seek the probability that the third beep occurs after one minute but not after three minutes, then your first method is correct. $$\mathsf P(1< T_3\leqslant 3)~{=\mathsf P(1<T_3)-\mathsf P(3<T_3) \\= \mathsf P(N_{[0;1)}\leq 2)-\mathsf P(N_{[0;3)}\leq 2)\\=(1+1+\tfrac 12)\mathsf e^{-1}+(1+3+\tfrac 92)\mathsf e^{-3}\\=\tfrac12(5\mathsf e^{-1}+17\mathsf e^{-3})}$$