Distribution of first arrival in Poisson Process Conditioned on that there's been only one arrival up until time t

Question

Consider a Poisson process with intensity λ > 0 with arrival times $T_1$, $T_2$, ...

Compute the distribution of the arrival time $T_1$ given that no more points arrived until time t (i.e. $N_t$ = 1).

Answer 1

Conditioned on $\{N_t=n\}$, the joint distribution of the arrivals $(T_1,\ldots,T_n)$ is that of the order statistics of the uniform distribution on $(0,t)$. That is, $T_i\stackrel d=U_{(i)}$ where $U_i\sim\mathrm{Unif}(0,t)$ and $U_{(i)}$ denotes the $i^{\mathrm{th}}$ order statistic.

Here $n=1$, so the distribution of $T_1$ conditioned on $\{N_t=1\}$ is very simple. Can you figure it out from here?