An armored car transports money between two locations and travels along one of three routes: \(A_{1}, B_{1}, C_{1}\). On one day, the route is randomly selected according to the probabilities \(P\left(A_{1}ight)=0.4, P\left(B_{1}ight)=0.2\), and \(P\left(C_{1}ight)=0.4\). However, on the second day, the probabilities change so that the probability of the previously chosen route is 0.2 with the other two routes equally probable. Denote those second-day routes by \(A_{2}, B_{2}\), and \(C_{2}\). Find \(P\left(A_{2} \mid A_{1}ight)\).