选the child playing video games 玩视频游戏的孩子.playing video games是定语.
即使the child's playing video games理解成所有格形式,也只能选后者,如果选前者,句子就成了The child's playing video games made his mother very angry.主语成了“playing video games ”,那his mother中的his作为主语所有格,就不是“孩子的”,反而成了“玩游戏的”了.