mersenneforum.org  

Go Back   mersenneforum.org > Fun Stuff > Puzzles

Reply
 
Thread Tools
Old 2008-04-11, 14:56   #12
cheesehead
 
cheesehead's Avatar
 
"Richard B. Woods"
Aug 2002
Wisconsin USA

22×3×641 Posts
Default

Quote:
Originally Posted by davieddy View Post
You swap. (You now know which of the two not to swap with!)
Pronoun correction to avoid potential confusion: [I]I[/I] swap. ([I]I[/I] now know which of the two not to swap with!)
cheesehead is offline   Reply With Quote
Old 2008-04-11, 16:06   #13
davieddy
 
davieddy's Avatar
 
"Lucan"
Dec 2006
England

2·3·13·83 Posts
Default

Quote:
Originally Posted by cheesehead View Post
Pronoun correction to avoid potential confusion: [I]I[/I] swap. ([I]I[/I] now know which of the two not to swap with!)
THX. I think I belatedly noticed this at the time. By "you" I meant "one".
I agree the confusion arose because I was previously using "you" to denote
the quizmasters


BTW I think we can safely dispense with the spoilers now

David

Last fiddled with by davieddy on 2008-04-11 at 16:35
davieddy is offline   Reply With Quote
Old 2008-04-20, 15:09   #14
davieddy
 
davieddy's Avatar
 
"Lucan"
Dec 2006
England

2×3×13×83 Posts
Default

I gave this to an old school friend of mine and he replied:

There are three scenarios:
I pick the door with the car, hosts reveals tricycle 1 or 2, I swap and lose the car.
I pick the door with tricycle 1, the hosts reveals tricycle 2, I swap and win the car.
I pick the door with tricycle 2, the hosts reveals tricycle 1, I swap and win the car.

So swapping is always worth it as it wins 2/3 of the time.

I think that puts it as clearly as possible.



davieddy is offline   Reply With Quote
Old 2008-04-21, 07:05   #15
Patrick123
 
Patrick123's Avatar
 
Jan 2006
JHB, South Africa

157 Posts
Default

Quote:
Originally Posted by davieddy View Post
There are three scenarios:
I pick the door with the car, hosts reveals tricycle 1 or 2, I swap and lose the car.
I pick the door with tricycle 1, the hosts reveals tricycle 2, I swap and win the car.
I pick the door with tricycle 2, the hosts reveals tricycle 1, I swap and win the car.

So swapping is always worth it as it wins 2/3 of the time.
Being devil's advocate...
It looks like 2 scenarios are summarized into one:
Should it not reflect:

I pick the door with the car, hosts reveals tricycle 1, I swap and lose the car.
I pick the door with the car, hosts reveals tricycle 2, I swap and lose the car.
I pick the door with tricycle 1, the hosts reveals tricycle 2, I swap and win the car.
I pick the door with tricycle 2, the hosts reveals tricycle 1, I swap and win the car.

Bringing it back to a 50/50 chance?

This is with the presumption that the hosts know what is behind each door and deliberately open a door that reveals a tricycle.

Regards
Patrick

Last fiddled with by Patrick123 on 2008-04-21 at 07:07 Reason: added the presumption.
Patrick123 is offline   Reply With Quote
Old 2008-04-21, 10:30   #16
davieddy
 
davieddy's Avatar
 
"Lucan"
Dec 2006
England

2×3×13×83 Posts
Default

Your 4 scenarios are not equally probable/frequent.
See my earlier posts.

David
davieddy is offline   Reply With Quote
Old 2008-04-21, 12:42   #17
Patrick123
 
Patrick123's Avatar
 
Jan 2006
JHB, South Africa

2358 Posts
Default

If it was a random selection that the hosts use to select the 'other door' to open then I would agree with you, but no matter which door you select, they will always ensure that they will open a door with a tricycle in it.

Thus what was initially a 1/3 probability, because they deliberately choose a door containing a tricycle no matter what door you choose, it becomes a standard 1/2 selection.
Patrick123 is offline   Reply With Quote
Old 2008-04-21, 13:35   #18
Patrick123
 
Patrick123's Avatar
 
Jan 2006
JHB, South Africa

157 Posts
Default

Allow me to extrapolate:

You choose a door - 66.7% chance it contains a tricycle.
They choose a door(at random or a member of the audience chooses it) 50% or 100% chance it contains a tricycle. if it does contain one then it becomes logical to switch to the other door(66.7% chance you've eliminated the tricycles), but because they know the outcome and do choose a door with a tricycle, you are still left with a 50% chance.

Last fiddled with by Patrick123 on 2008-04-21 at 13:47 Reason: member of audience edit.
Patrick123 is offline   Reply With Quote
Old 2008-04-21, 19:40   #19
rogue
 
rogue's Avatar
 
"Mark"
Apr 2003
Between here and the

3·1,973 Posts
Default

Quote:
Originally Posted by Patrick123 View Post
If it was a random selection that the hosts use to select the 'other door' to open then I would agree with you, but no matter which door you select, they will always ensure that they will open a door with a tricycle in it.

Thus what was initially a 1/3 probability, because they deliberately choose a door containing a tricycle no matter what door you choose, it becomes a standard 1/2 selection.
I think you misunderstand. The probability of the host selecting the door with the tricycle doesn't matter. It is the probability of which of the remaining doors has a tricycle. Just because it is a 100% chance that the host will choose an unopened door with the tricycle doesn't affect the probability that there is a 2/3 chance if there being a tricycle there in the first place.
rogue is offline   Reply With Quote
Old 2008-04-21, 19:58   #20
davieddy
 
davieddy's Avatar
 
"Lucan"
Dec 2006
England

2×3×13×83 Posts
Default

Quote:
Originally Posted by Patrick123 View Post
If it was a random selection that the hosts use to select the 'other door' to open then I would agree with you, but no matter which door you select, they will always ensure that they will open a door with a tricycle in it.
On the contrary. If the host picks a door at random, it may be
the car (and you presumably lose it). But if he reveals a tricycle and
offers you the swap, NOW it is a 50/50 proposition.

(Why?)

David
davieddy is offline   Reply With Quote
Old 2008-04-22, 04:56   #21
robo_mojo
 
robo_mojo's Avatar
 
Mar 2008

1000002 Posts
Default

Quote:
Originally Posted by Patrick123 View Post
Being devil's advocate...
It looks like 2 scenarios are summarized into one:
Should it not reflect:

I pick the door with the car, hosts reveals tricycle 1, I swap and lose the car.
I pick the door with the car, hosts reveals tricycle 2, I swap and lose the car.
I pick the door with tricycle 1, the hosts reveals tricycle 2, I swap and win the car.
I pick the door with tricycle 2, the hosts reveals tricycle 1, I swap and win the car.

Bringing it back to a 50/50 chance?

This is with the presumption that the hosts know what is behind each door and deliberately open a door that reveals a tricycle.

Regards
Patrick
By supposing those four separate scenarios are equally probable, you are supposing a 50% chance that the contestant chose the winning door to begin with, when it is only 1/3. Therefore your scenarios cannot be equally probable. Your four scenarios should be weighted like this:

(1/6) I pick the door with the car, hosts reveals tricycle 1, I swap and lose the car.
(1/6) I pick the door with the car, hosts reveals tricycle 2, I swap and lose the car.
(1/3) I pick the door with tricycle 1, the hosts reveals tricycle 2, I swap and win the car.
(1/3) I pick the door with tricycle 2, the hosts reveals tricycle 1, I swap and win the car.

If you don't understand why the first two scenarios must be 1/6, then consider that Monty's selection is only made after the contestant picks a door. If the contestant picks a winning door (with 1/3 prob), then the odds that Monty will pick each of the two wrong doors is 1/3 * 1/2 or 1/6. In this case it also really doesn't matter which one he picks, so it is simpler to roll them into one case where the contestant has picked the right door first.

Last fiddled with by robo_mojo on 2008-04-22 at 05:21
robo_mojo is offline   Reply With Quote
Old 2008-04-22, 05:20   #22
robo_mojo
 
robo_mojo's Avatar
 
Mar 2008

2016 Posts
Default

Quote:
Originally Posted by davieddy View Post
On the contrary. If the host picks a door at random, it may be
the car (and you presumably lose it). But if he reveals a tricycle and
offers you the swap, NOW it is a 50/50 proposition.

(Why?)

David
I hope it is OK for me to answer:

Because, suppose you have a 1/3 chance of picking the car at first. If the host COULD reveal any door, even a car, and have you lose it, then we cannot use his limits to increase our odds. So we have a 1/6 (1/3 * 1/2) chance that we had the car first, kept it and won; a 1/6 (1/3 * 1/2) chance we had the car first, but swapped and lost; 1/6 (2/3 * 1/2 * 1/2) chance we picked a wrong door and the host reveals a wrong door, and we kept the door and lost; a 1/6 (2/3 * 1/2 * 1/2) chance we had a wrong door, the host reveals a wrong door, we swap and win; and a 1/3 (2/3 * 1/2) chance that we picked a wrong door, and the host reveals the car, preventing us from having the opportunity to swap. Then, supposing we have an opportunity to swap, removing the last outcome (assuming the host hadn't revealed the car, though he COULD have), then there's a 1/2 chance that the swap will win and a 1/2 chance a swap would lose. Overall in that kind of game you'd still have only a total of 1/3 chance of winning the car.

Again that's supposing that the host is NOT limited to picking a wrong door.

Only when the host is limited to picking a wrong door can you increase your odds to 2/3 by swapping, as was already discussed.

Where most people get confused about this kind of problem is that you must figure out how the outcomes are dependant of each other (what the restrictions are), and multiply the odds of each element together in the chain.

Last fiddled with by robo_mojo on 2008-04-22 at 05:23
robo_mojo is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Titan's Best Choice Brain GPU Computing 30 2019-10-19 19:19
Disobedience is a choice jasong jasong 21 2018-02-11 00:55
Hardware purchase choice for NFS? VBCurtis Factoring 4 2013-11-15 07:22
GIMPS' charity of choice ixfd64 Lounge 0 2010-03-02 22:00
On the skew choice for SNFS Batalov Factoring 1 2009-11-17 21:18

All times are UTC. The time now is 01:35.

Tue Sep 29 01:35:09 UTC 2020 up 18 days, 22:46, 0 users, load averages: 1.37, 1.60, 1.51

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.