 Solution

After the third arrival, the future interarrival times are independent of the first three interarrival times and again generate a rate-$\lambda$ Poisson process. Conditional on $T_3=t<2$, exactly three arrivals have occurred by time $t$, and the number of further arrivals before two is Poisson with mean $\lambda (2-t)$. Consequently, in the regular conditional-distribution sense at $t=1$,Explicitly its conditional mass is zero for $k<3$ and isThis conditions on the third arrival time itself, rather than merely on the event that the third arrival has happened by time one.