We model the time-resolved and time-integrated photoluminescence of a single InAs/GaAs quantum dot (QD) using a random population description. We reproduce the joint power dependence of the single QD exciton complexes (neutral exciton, neutral biexciton and charged trions). We use the model to investigate the selective optical pumping phenomenon, a predominance of the negative trion observed when the optical excitation is resonant to a non-intentional impurity level. Our experiments and simulations determine that the negative charge confined in the QD after exciting resonance to the impurity level escapes in 10 ns.