I agree we need to be clear that Monty must open a door, that the door Monty opens must hide a goat, and I must be given the option to change my choice. What I don't understand is that some say it is also required that when I originally choose the door hiding the car, such that Monty has his choice of two goat doors to open, that Monty must make this choice randomly.
It seems to me it does not make a difference whether Monty chooses randomly or not.
Suppose Monty has fallen into the pattern that when I have already chosen the door with the car, that Monty will always open the goat door "to the immediate circular right" of my chosen door. (Car behind door 1 and I choose 1, then open door 2. Car behind door 2 and I choose 2, then open door 3. Car behind door 3 and I choose 3, then open door 1.)
Now suppose I don't know Monty has fallen into this pattern, so I always switch. I still win two-thirds of the time.
Suppose I know Monty has fallen into this pattern, but I still always switch. I still win two-thirds of the time.
Suppose I have watched the show many times and have figured out that Monty always picks the door to the immediate circular right in those instances when I have already picked the car. Because I know this, I always hold my original choice when he opens the door to the immediate right of my choice, and I always switch when he opens the door that's not to the immediate right of my choice. If I worked out the options correctly, this strategy yields a different set of wins and losses, but it still turns out that I win two-thirds of the time.
What if Monty's bias is not 100%. Let's say 75% of the time he chooses the goat immediately right of my car choice, but 25% of the time he makes the other choice. In this case, if I use my knowledge of his bias and I always hold my choice when he opens the door to the right of my choice, then I win only 64.6 % of the time by trying to use my knowledge of his bias, but I could still win 66.7% of the time by always switching.
If his bias to the right is not so strong, and approaches random (50/50) then holding whenever he opens the door to my right will only result in 50% wins.
Therefore, it seems to me I should still continue the pattern of always switching even though Monty might have some bias in how he chooses between the two goats when he has two goats to choose from. Always switching still yields a two-thirds win pattern, but trying to consider Monty's bias only yields a two-thirds win pattern if his bias is 100%. If his bias is less than 100%, then trying to second-guess him will yield less than two-thirds win pattern. So don't try to deal with any bias Monty might have... always switch.
What am I missing? Why do some people make the specification that Monty must choose randomly?
Michael D. Edmiston, PhD.
Professor of Chemistry and Physics
Chair, Division of Natural and Applied Sciences
Bluffton University
Bluffton, OH 45817
Office 419-358-3270
Cell 419-230-9657