The problem is very well described and explained by a plenitude of sources. So, just a very brief reminder:
a guest is given the opportunity to select one of the three closed doors. There is a prize behind exactly one of the doors.
Once the guest has selected a door, the host opens one another (not selected) door to reveal that it hides nothing.
Then the guest can either confirm the initial choice or select another (obviously still closed) door.
Which strategy leads to success with higher probability?
Long story short, if the guest's decision remains unchanged then the probability is 1/3, if the guest changes the selection then the probability is 2/3
The solution can be found elsewhere, along with the software simulators (among them one is mine https://github.com/tms320c/threedoorstrial)
The solution itself is fine no doubt, but let us change the trial a bit.
After the guest has made the first selection, the host removes the open box (let's use boxes instead of the doors). Because everybody knows that the open box contains nothing this action does not change the distribution of the probabilities (or does it? see the questions below).
Now we have two boxes, which are not equal: one contains the prize with the probability of 1/3, while the probability to find the prize in another box is 2/3.
The host says goodbye to the first guest and invites another one. The new guest is given the opportunity to select one of the two closed boxes.
Because new guest does not know the history, so the assumption is that the probability to get the prize is 1/2. Which is totally wrong as we all (and the host, and the first guest) know now.
If the host is a kind of generous person he can tell the story. After that, the new guest can get the prize with 2/3 probability. Is it a correct probability?
I propose to discuss the following questions:
if you are in a situation where you should choose one of two presumably equal boxes/packages/whatever, should you insist on the disclosure of their history? Perhaps, they were part in such trial once upon a time in a galaxy far far away and are not equal in fact.
for how long the boxes are in the non-equilibrium state? The time delay between the two trials (first and second guests) can be arbitrary long.
may the situation lead to the conclusion that probability depends on someone's opinion, and your decision always explicitly depends on someone else's decision, which could be made in a deep past?
may the probability to have a treasure be an internal property of every box that can be modified mentally by humans?
Modify the problem to have 100 doors. The contestant chooses one. Monty then opens 98 doors (not including the chosen one) that he knows do not hide the prize. The contestant's first choice had 1% chance of being correct. Switching doors will always be advantageous in this case, since now the remaining unchosen door has a 99% chance of hiding the prize.
Bringing in a second contestant who has no knowledge of previous choices will always have a 50% chance of choosing the correct door. The first contestant's advantage comes as a result of his knowledge of his first choice. If the second contestant is given knowledge of the first contestant's choice (or merely asked if he wants to switch doors based on previous contestant's choice), the 99% chance of winning is preserved by switching.
The problem is in fact simple: you describe somebody's mind interacting with an "objective" environment: probability generally describes this interaction. Thus, if you change the "mind" conditions you change probabilites. At abstract level, this corresponds with the subjective probability interpretation of Savage. As a sort of self-advertisement I point to the paper
@article{0367271,
author={M. K{\'{a}}rn{\'{y}} and T. Kroupa},
title={Axiomatisation of fully probabilistic design},
journal={Inf. Sci.},
number={1},
pages={105-113},
volume={186},
publisher={Elsevier},
year={2012},
}
which simply interpretes probability as the inevitable tool for decsion making under uncertainty.
Good news is that the subjective probability can be corrected by sufficiently informative observed data. See e.g.
@incollection{BerKar:97,
author="L. Berec and M. K{\'a}rn{\'y}",
title="Identification of reality in {B}ayesian context",
booktitle={Computer-Intensive Methods in Control and
Signal Processing},
publisher={Birkh\"auser},
pages="181--193",
year=1997,
editor="K. Warwick and M. K{\'a}rn{\'y}"
}
where you will find explicit influence of the considered model set and the knowledge stored ...
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