Answer
$3\;eV$
Work Step by Step
The simulated emission occurs when atoms at a higher energy level having apparently long life time are deexcited to a lower energy level having apparently short life time due to simulation of external photon having equivalent energy of the energy difference between above two levels.
The light emitted by $A$ illuminates $B$ and can cause stimulated emission of $B$. Thus, the energy of the emitted photon from $A$ should be equivalent to the energy difference between the $B$'s energy levels which participates in stimulated emission.
To satisfy the above two conditions, the transition in $A$ should be from $6.9\;eV$ (lifetime: $3\;ms$) to $3.9\;eV$ (lifetime: $3\;\mu s$) and the transition in $B$ should be from $10.8\;eV$ (lifetime: $3\;ms$) to $7.8\;eV$ (lifetime: $3\;\mu s$).
Therefore, the energy per photon of that stimulated emission of $B$ is $(6.9-3.9)\;eV=3\;eV$.
