As a wrongologist, I was inspired by Andy Brodie's Monty Hall problem blog to ponder a little about why the problem causes so much debate.
The thought procedure that people apply instinctively is this: there are 3 doors, and the contestant has a random choice, therefore the odds are 1/3 for each door. Even after one door is opened, it makes no difference whether you stick or change, the remaining 2 doors you can choose from are equally likely so, as one door has been removed, the odds are now 50:50.[1]
These statements are correct and the logical deduction (as stated) is correct too, but the answer is (sorry) not. What's missing is that this analysis is incomplete. The contestant makes random choices, yes, but there is another actor in the scenario. In an entirely random world, where the presenter also made a random choice, the odds for the last door would stay at 1/3, and between the remaining two doors they would effectively be 50%[1] but some of the time the presenter would open the door which hides the cadillac.
In actuality, the presenter deliberately chooses not to open a door hiding a cadillac, and so introduces a non-random factor which is why the instinctive answer is not the right one after all.
When the contestant makes the original decision, the odds were 1/3 for the chosen door, and 2/3 for the remaining two doors. But when the presenter opens a door, this guarantees that the contestant knows for certain what is behind the open door. And by implication, this has shared information about what's behind the unopened door. In other words, the contestant knows that - of the original 2/3 case - the cadillac will be behind the 3rd (unchosen) door 100% of the time. 100% of 2/3 is: 2/3.
The odds for the first door haven't changed, they are still 1/3rd. 100% of 1/3rd. But now you're comparing a 2/3 choice with a 1/3 choice, and that is not equal at all.
So the verdict is in: you should opt to change your mind and pick a door that you originally rejected, which the presenter also rejected. Because while your rejection was made at random, his was not: he knew something when he rejected that door.
Or to put it another way, your facts were correct, your logic was correct, and you may still have come up with the wrong answer because you missed a factor. I work in an environment where wrongness is punished so heavily that if you make an incorrect deduction, you are considered to have used up your question-asking tokens for the day, or possibly the week or the month.
It's a scary world out there, full of pedants, perfectionists, and a loud get-it-right-first-time brigade. Wrongologists of the world, join me in standing up for your wrong deductions. When you go away and muse over the most important question: where you went astray, you will learn more than if you had originally come to the right answer quickly. I also believe that getting things wrong and understanding why is an important step in understanding a customer point of view and how easy (or otherwise) our output is to consume. Wrongologists, welcome to your new role as consumability champions.
[1] I know, I know, I'm mixing percentages, ratios, and fractions. Life is more exciting if you live on the edge.
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