Monty Hall With A Twist

Most people by now have heard of the Monty Hall problem which is a problem of logic and probability whose correct answer is somewhat counter-intuitive. In this post, I present an extension to the Monty Hall problem which I am calling Monty Hall With A Twist.

The Monty Hall problem...

Skip down to see the new problem, this section serves to restate the original Monty Hall problem and its solution.

To recap: There exists a game show with a host ("Monty Hall") and a contestant. There are 3 closed doors and it is known that behind one of the doors there is a prize (represented by a car) and behind each of the two others is a single goat. The contestant is asked to choose a door, without opening it. The host then opens one of the other two doors and a goat is always revealed (either by design or by accident) - the door containing the car is never opened, at this stage. The host then offers the contestant the chance to stick with his original choice or switch to the other closed door. The door the contestant finally settles on is opened and if the car is behind that door, she gets to keep it.

The question is: should the contestant stick or switch?

It turns out that the correct answer to this problem is that the contestant should switch because 2/3 of the time, the car will be behind the other door.

There are many explanations about why this is true, but the one I like is the following. When the contestant makes the original choice, the original selection has 1/3 a chance of being the correct one (all other things being equal). This means that the group of the other 2 doors had a 2/3 chance of containing the door with the car. This remains true when one of the doors in that group is opened - that group of 2 doors originally had a chance of 2/3 of containing the car and it still does, but now we know that one of those 2 doors contains a goat and we know which of those 2 doors that is. So the total probability of the group of 2 doors hiding a car is unchanged, but it can't be the one with the goat, which means that the remaining door must have a 2/3 chance of being the one with the car - hence the contestant should switch.

...with the twist

So that's the original Monty Hall problem. Here it is again with a twist.

We are going take the original problem and make some additional stipulations:

• the host knows which door conceals the car
• if the host has a choice of action, then his contract with the TV network states that he must choose the choice that minimises the chance that the contestant will win the prize
• after the host opens the first door containing a goat, the host MAY choose to explain the logic of the solution to the original Monty Hall problem to the contestant. His choice must be guided by the second stipulation
• the host is unaware if the contestant already knows the correct solution to the Monty Hall problem
• the contestant is aware of these stipulations, even if she does not know what the Monty Hall problem is or what its correct solution is.

In the case that the host chooses not to explain the logic of Monty Hall problem to a contestant who happens to know the correct solution to the Monty Hall problem anyway, should the contestant stick or switch?