A Monty Puzzle

A Monty Puzzle

You are a contestant on a game show.

Here is the setup:

The host takes the contestant into a room and shows him 2 boxes.

On top of one box is a card that reads "A"
On top of the other box is a card that reads "B"

The host tells the contestant that:

In one box is a wonderful prize.
In the other box is a dud prize.

Before he came into the room a lovely mystery assistant
has chosen box "A" for him.

The host asks the contestant if he would like to stick with the
assistant's choice or switch to box "B"...

Questions:

Can what happened in the room before the contestant entered
affect the contestants chance of winning the prize?

Should the contestant stick with the box chosen for him or switch?

Does the contestant's chance of winning depend on whether the
host is telling the truth about which box the assistant
chose for the contestant?

Does the contestant's chance of winning depend on how the assistant
chose the box for him?

Explain/discuss what you would do and why...