Monty Hall and the 1000 Doors

Logic Level 1

Monty Hall decides to perform his final episode of the Let's Make a Deal series with a little twist, and Calvin is elated when he is the first contestant to be called to the stage. Everyone eagerly watches as Calvin approaches the stage, waiting to see the surprise Mr. Hall will present.

Mr. Hall then hollers into his microphone, "AND THE SURPRISE IS...... CALVIN WILL BE PLAYING WITH 1000 DOORS INSTEAD OF JUST 3." The audience gasps, but Mr. Hall keeps talking, explaining the rules:

999 of the 1000 doors covers a pig, while the last covers the most amazing super special magnificent supercar of your dreams .
Calvin will pick 1 of the 1000 doors, after which I will open 998 doors which cover a pig, leaving two doors for Calvin to choose from.
Calvin will then have to choose whether to open the door he originally chose or the door I left closed.

Should Calvin open the door which he originally chose, or switch to the door which Mr. Hall did not open?

Calvin should switch, the probability of winning is

50.1\%

Calvin should switch, the probability of winning is

99.9\%

Calvin should not switch, the probability of winning is a little over

50\%

Calvin should not switch, the probability of winning is

75.0\%

The probability of winning is

50\%

regardless

2 solutions

Infinity Mathematics
May 24, 2018

When Calvin chooses whichever door he believes holds the supercar, there is a $\frac{1}{1000}$ probability that this door actually does have the supercar behind it, while there is a $\frac{999}{1000}$ probability that any of the other 999 doors.

Assuming (with a probability of $\frac{999}{1000}$ ) that one of these 999 doors does have the supercar, we can divide the 1000 original doors into two sets, the door which Calvin chose and the 999 doors which were not chosen. Among these 999 doors, if Mr. Hall opens 998 doors which cover pigs, then (since we assumed that one of the 999 doors has the supercar) the final door that is left closed must obviously hold the supercar. However, since there is a $\frac{999}{1000}$ chance that the other 999 doors actually has the supercar (there is still a $\frac{1}{1000}$ chance that the door Calvin chose had the car), there is a $\frac{999}{1000}$ chance that the door left closed by Mr. Hall has the supercar.

So, Calvin will almost certainly, with a probability of 99.9% or 999/1000, choose the supercar if he chooses the door which Mr. Hall leaves unopened. Maybe playing with 1000 doors wasn't that bad after all.

Michael Mendrin
May 24, 2018

This makes no sense! Let's say that I've taken my unpublished book to a publisher, and they offer me meager compensation for it. So, I try 998 other publishers, and they all reject me outright. Should I then try the remaining 1000th publisher, or should I just stay with the first one? My feeling would tell me that if 998 others have rejected me, the last one is REALLY gonna!

Your example differs from the question, in that it is a certainty that 1 of the 1000 is good, and all the rest are bad

Stephen Mellor - 3 years ago

Too bad it doesn't work that way in the publishing world, huh

Let's say that Monty Hall is a bit inebriated on that day, and he's not even sure if there's even a car behind any of the doors and doesn't know where it'd be even if there was one. So, he opens 998 doors at random and there's no car. What's the probability that behind the 1000th door there is a car? Let's say the probability that there is a car at all behind any door is $P$ . I know this is a different kind of problem, and meanwhile I totally agree with your posted analysis of the 1000 door Monty Hall problem.

Michael Mendrin - 3 years ago

P if multiple cars are allowed (doors are i.i.d.), otherwise you have 1000 possibilities + 1 (no car at all) with P = 1/1001 if it’s uniform. You’re initial choice is 1/1001, but after opening 998 doors, it is 1/3 for the remaining door. So you should switch again. But too often humans just stick with their past decisions because they want to be self consistent.

Michael B - 2 years, 10 months ago

In this problem, it is certain that one of the doors holds the supercar, while your example is a case where uncertainty is prevalent no matter what.

Infinity Mathematics - 3 years ago

Yeah I know that.

Michael Mendrin - 3 years ago

...then why did you use that example?

Louis Ullman - 3 years ago

@Louis Ullman – Oh probably because I've used the exact same argument Infinity Mathematics gave here with some of my friends in the past which I've tried to explain to about the Monty Hall problem, and that's the kind of thing I keep hearing from them. It never fails to amaze me how difficult it can be to convince people that the best choice is to swtich, once they've made a decision.

I was just playing stupid, okay?

Michael Mendrin - 3 years ago

it‘s because the only move wich can change the outcome of the game is the first one, if you choose the right one, you lose if you choose the wrong one, you win, so it’s 1 door against 999. because all other wrong ones get eliminated.

Paris Saliveros - 3 years ago

yes you're right

Michael Mendrin - 3 years ago

The correlation with Monty Hall can apply when there is only 1 of an entire bunch of similar possibilities that leads to a 'favorable' outcome. What you describe is not a similar problem.

Rama Ganesan - 2 years, 9 months ago

Monty Hall and the 1000 Doors

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