Additionally, what was the likelihood of a commoner rising into the bureaucracy through the exams?
By no means discouraging other answers, but there was quite a lively discussion of the topic here, including an answer by yours truly: https://www.reddit.com/r/AskHistorians/comments/3cvk90/how_effective_were_the_meritocracies_of_imperial/
I also had a post on Imperial Chinese education make it into the Sunday Digest; you can read it here: https://www.reddit.com/r/AskHistorians/comments/3o64f2/im_a_son_to_a_wealthy_chinese_noble_in_the_17th/cvvzgug