## Pick the Door with the Prize!

What is the basis for reason? And mathematics?

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Mike Strand
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### Re: Pick the Door with the Prize!

Thank you, wtf, for joining in and giving a clear explanation of the odds involved in this game:
wtf wrote: Tue Jul 03, 2018 9:33 pm The problem becomes clear once we explicitly state the assumptions.

* Monty knows what's behind each door.

* Monty must reveal exactly one door.

* Monty must reveal a door concealing a goat.

Now the puzzle is clear.

The player makes a choice.

1/3 of the time the player chooses correctly and the other two doors conceal goats. Monty reveals one of the goats. Switching picks the other goat and the player loses.

The other 2/3 of the time, the player has initially chosen a goat. There are two doors left, one concealing the car and the other concealing a goat. Monty must reveal the door concealing the goat. The remaining door contains the car. Switching wins the car.

So 1/3 of the time switching loses; and 2/3 of the time switching wins.

Let me repeat this. 2/3 of the time the player's initial guess is a door with a goat. That leaves one door with a car and one with a goat. Monty is required by the rules of the game to reveal the door with the goat. That means that switching guarantees we get the car. This case happens 2 times out of 3.

1 time out of 3 the player's initial guess is the door with the car. Switching loses.

So switching is the winning strategy in 2 out of 3 equally likely cases.
Any reactions from other participants?

I hope someone now will answer the question of how much a gambling casino should charge a person to play the game. After that, I hope you'll be interested in looking at or presenting generalizations of the game: What the player should do to maximize the odds of winning, if there are more than three doors, and also if Monty leaves more than two unopened doors, one of which is the player's first choice.

To wtf: Posing the game with goats as losers and a car as the prize led to this anecdote: There was a farmer who played the game and never switched, because he wanted a goat more than the car.
wtf
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### Re: Pick the Door with the Prize!

Mike Strand wrote: Tue Jul 03, 2018 11:39 pm To wtf: Posing the game with goats as losers and a car as the prize led to this anecdote: There was a farmer who played the game and never switched, because he wanted a goat more than the car.
There's an xkcd for that.

https://xkcd.com/1282/
Mike Strand
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### Re: Pick the Door with the Prize!

wtf wrote: Wed Jul 04, 2018 12:25 am
Mike Strand wrote: Tue Jul 03, 2018 11:39 pm To wtf: Posing the game with goats as losers and a car as the prize led to this anecdote: There was a farmer who played the game and never switched, because he wanted a goat more than the car.
There's an xkcd for that.

https://xkcd.com/1282/
Thanks, wtf, for this link! I thought I was being original ...
mickthinks
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### Re: Pick the Door with the Prize!

Mike Strand
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### Re: Pick the Door with the Prize!

mickthinks wrote: Wed Jul 04, 2018 3:37 pm search.php?keywords=monty+hall

Just saying ...

I was joking about the fact that the author of the xkcd cartoon (in the link provided by mtf) and I had a similar thought about goats in the Monty Hall problem (coincidence). Thanks to mtf for introducing me to xkcd!

Although the Monty Hall three-door problem is well-known, it still defies the intuition of many folks, and it's fun to wrestle with it. That's why I started a thread about it -- to review it and to look at generalizations of it. I guess one of the philosophical aspects of this has to do with intuition -- what is it, and can it be trained for improvement?

The training possibility is the reason I suggested simulating the game, in earlier posts in this thread. Otherwise, this is just a specific example of the application of mathematical logic. Also, it's an interesting example of how logic can be at odds with intuition in some cases.

Here's the general problem: Given N doors, where N is at least 3, and exactly one door conceals the prize and is known only to the game host. Player picks one door as the prize door. Host opens X doors not containing the prize other than the door the player picked first, where X is less than or equal to N-2.

1. What is the probability of winning if the player doesn't switch doors?

2. What is the probability of winning if the player switches
(a) when X=N-2?
(b) when X is less than N-2 but greater than or equal to 1, and the player gives equal probability to the doors other than his first pick?

If you want to practice first, start with N=4 and X=2, then X=1. Of course, with no switch, the probability of winning is 1/4.
mickthinks
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### Re: Pick the Door with the Prize!

I think you have misunderstood the significance of the long list of hits when searching for "Monty Hall" here at the Phorum. If you look at the earliest result returned, you'll find it is this post from way back in 2012. It was posted by someone you know rather well.

lol
Mike Strand
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### Re: Pick the Door with the Prize!

mickthinks wrote: Wed Jul 04, 2018 10:06 pm I think you have misunderstood the significance of the long list of hits when searching for "Monty Hall" here at the Phorum. If you look at the earliest result returned, you'll find it is this post from way back in 2012. It was posted by someone you know rather well.

lol
Yes, I posted on this several years back, and I'm LOLing along with you, mickthinks. The Monty Hall game is a favorite of mine, and I wanted to revisit it after several years absence from this forum. And perhaps, several years absense of ... my mind or memory.

One of the reasons I like the three-door thing is because of intution vs. logic. I suggested simulations as a way to train the intuition in such cases.

Even though my training is in math, particularly probability and statistics, my intuition is often at odds with the results of logical demonstrations. Maybe others don't suffer as much from this pesky malady. So I like to revisit things i've been away from for years since retiring, and I apologize to those for which Monty Hall is uninteresting. For another example, non-Euclidean geometry and its application to advanced physics often defies my intuition, even in cases where I can follow the logic. Euclidean geometry is tough enough for me.

Thanks again for your interest, mickthinks (from mikeforgets).
Mike Strand
Posts: 406
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Location: USA

### Re: Pick the Door with the Prize!

Mike Strand wrote: Tue Jul 03, 2018 11:39 pm I hope someone now will answer the question of how much a gambling casino should charge a person to play the game. After that, I hope you'll be interested in looking at or presenting generalizations of the game: What the player should do to maximize the odds of winning, if there are more than three doors, and also if Monty leaves more than two unopened doors, one of which is the player's first choice.
Mike Strand wrote: Wed Jul 04, 2018 5:01 pm Here's the general problem: Given N doors, where N is at least 3, and exactly one door conceals the prize and is known only to the game host. Player picks one door as the prize door. Host opens X doors not containing the prize other than the door the player picked first, where X is less than or equal to N-2.

1. What is the probability of winning if the player doesn't switch doors?

2. What is the probability of winning if the player switches
(a) when X=N-2?
(b) when X is less than N-2 but greater than or equal to 1, and the player gives equal probability to the doors other than his first pick?

If you want to practice first, start with N=4 and X=2, then X=1. Of course, with no switch, the probability of winning is 1/4.
Is anyone still interested in these two exercises?

You may give qualitative answers if you wish. For example, in the general problem of N doors, does it improve the chance of winning to switch, if Monty opens at least one of the doors other than the one the player picked? Do the odds improve as the number of doors increases to N-2?
wtf
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### Re: Pick the Door with the Prize!

Mike Strand wrote: Sun Jul 08, 2018 7:18 pm Is anyone still interested in these two exercises?
You should take a look at the infinite hat puzzle. Much more counterintuitive.

https://en.wikipedia.org/wiki/Hat_puzzle
Mike Strand
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Location: USA

### Re: Pick the Door with the Prize!

wtf wrote: Sun Jul 08, 2018 7:38 pm
Mike Strand wrote: Sun Jul 08, 2018 7:18 pm Is anyone still interested in these two exercises?
You should take a look at the infinite hat puzzle. Much more counterintuitive.

https://en.wikipedia.org/wiki/Hat_puzzle
Looks like fun, wtf, and thanks for the link. Maybe you should start a thread on it.
wtf
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### Re: Pick the Door with the Prize!

Mike Strand wrote: Sun Jul 08, 2018 10:30 pm
Looks like fun, wtf, and thanks for the link. Maybe you should start a thread on it.
Phil is the thread starter here.