Is it permissible to profit from watching promotional YouTube videos via a website that includes music and entertainment scenes, knowing that the viewer mutes the sound and avoids what is forbidden, by entering a daily draw for random numbers to win money if they win?

There is no objection to earning money from watching videos if they are permissible and free from শরীয়াহ prohibitions, and if the wage and method of earning are known and free from gharar (excessive uncertainty) and gambling. However, if the compensation for watching is obtained through a draw, which the viewer may or may not win, then this is not permissible শরীয়াহ-wise; because it is gharar resembling gambling, and because the wage in a lease (ijarah) is stipulated to be known.