We explored how robot-assisted therapy based on smile analysis may facilitate the prosocial behaviors of children with autism spectrum disorder. Prosocial behaviors, which are actions for the benefit of others, are required to belong to society and increase the quality of life. As smiling is a candidate for predicting prosocial behaviors in robot-assisted therapy, we measured smiles by annotating behaviors that were recorded with video cameras and by classifying facial muscle activities recorded with a wearable device. While interacting with a robot, the participants experienced two situations where participants' prosocial behaviors are expected, which were supporting the robot to walk and helping the robot from falling. We first explored the overall smiles at specific timings and prosocial behaviors. Then, we explored the smiles triggered by a robot and behavior changes before engaging in prosocial behaviors. The results show that the specific timing of smiles and prosocial behaviors increased in the second session of children with autism spectrum disorder. Additionally, a smile was followed by a series of behaviors before prosocial behavior. With a proposed Bayesian model, smiling, or heading predicted prosocial behaviors with higher accuracy compared to other variables. Particularly, voluntary prosocial behaviors were observed after smiling. The findings of this exploratory study imply that smiles might be a signal of prosocial behaviors. We also suggest a probabilistic model for predicting prosocial behaviors based on smile analysis, which could be applied to personalized robot-assisted therapy by controlling a robot's movements to arouse smiles and increase the probability that a child with autism spectrum disorder will engage in prosocial behaviors.