  :You still have not got it. The simple symmetrical solution does address the question 'which strategy is best in a specific case such as that of a player who has picked Door 1 and has then seen the host open Door 3?'.  If the host action is random, it does not matter that the host has opened a specific door, because we know that the door opened makes no difference, by reason of symmetry.  Can any one else explain this better?   