Puzzle - which door?

[XTR]

just read this thread and my head is baffled!!!!!!!!


i just dont get it... dont try and explain it to me cause im baffled enough......

[Zathras]

Thats exactly what I'm saying. Because the host will only ever open a non winning door, your initial choice has no bearing on the actual chance of winning.

Yes it does! If you choose a wrong door, probability 2/3 as there's two wrong, one right, the host HAS TO remove the other non-winning door, therefore if you switch you DEFINITELY get the prize. Your initial choice does affect the probability at the second stage. If you stick, you definitely lose.

With me so far?

Probability of picking a wrong door first = 2/3
If you switch, probability of picking a wrong door then winning = 2/3
If you stick, probability of picking a wrong door then winning = 0

If you choose the right door, probability 1/3, and the host removes a wrong door, if you switch you definitely lose but if you stick you definitely win.

Probability of picking a right door first = 1/3
If you switch, probability of picking a right door then winning = 0
If you stick, probability of picking a right door first = 1/3

So, to sum up, your chance of winning when switching is 2/3
Your chance of winning when sticking is 1/3

You get extra information after your first pick as a wrong door is definitely removed. If you don't use this information, it's as if a wrong door was never opened, therefore you have a one in three chance of choosing the right door. If you do use this information, it doubles your chances.

[Nox]

its easy to get your head round it. look at it this way:

there are 1000 doors. you have a 1 in a thousand chance of guessing the right door.

you guess a door, think of it as your 1 in a thousand chance of winning.

now, think of the other 999 as one door with a 999 in a thousand chance of winning.

switching to the other group of doors is like giving you 999 doors for the price of one, because the host will remove 998 WRONG ones from your choice.

got it?

Nox

[Rythmic]

OK - I spent some time in excel. Heres a table containing all eventualities - if I've missed one, please let me know:

Prize	Player 	Host	Player 	Player 	Player 	no	change
Is	Initially 	Reveals	Finally	wins	changed	change	and
Behind	picks:		picks		choice	and win	win
1	1	2	1	TRUE	FALSE	FALSE	TRUE
1	1	3	1	TRUE	FALSE	FALSE	TRUE
1	1	2	3	FALSE	TRUE	FALSE	FALSE
1	1	3	2	FALSE	TRUE	FALSE	FALSE
1	2	3	2	FALSE	FALSE	FALSE	FALSE
1	2	3	1	TRUE	TRUE	TRUE	FALSE
1	3	2	3	FALSE	FALSE	FALSE	FALSE
1	3	2	1	TRUE	TRUE	TRUE	FALSE
2	1	3	1	FALSE	FALSE	FALSE	FALSE
2	1	3	2	TRUE	TRUE	TRUE	FALSE
2	2	1	2	TRUE	FALSE	FALSE	TRUE
2	2	3	2	TRUE	FALSE	FALSE	TRUE
2	2	1	3	FALSE	TRUE	FALSE	FALSE
2	2	3	1	FALSE	TRUE	FALSE	FALSE
2	3	1	3	FALSE	FALSE	FALSE	FALSE
2	3	1	2	TRUE	TRUE	TRUE	FALSE
3	1	2	1	FALSE	FALSE	FALSE	FALSE
3	1	2	3	TRUE	TRUE	TRUE	FALSE
3	2	1	2	FALSE	FALSE	FALSE	FALSE
3	2	1	3	TRUE	TRUE	TRUE	FALSE
3	3	1	3	TRUE	FALSE	FALSE	TRUE
3	3	2	3	TRUE	FALSE	FALSE	TRUE
3	3	1	2	FALSE	TRUE	FALSE	FALSE
3	3	2	1	FALSE	TRUE	FALSE	FALSE

As you can see - there are 24 possible paths

12 result in a win - thats 12/24 or a 50% chance to win

In 12 paths the player changed their choice - of these 6 resulted in a win - 6/12 means a 50% chance of winning if you change your choice

In 12 paths the player didn't change their choice - of these 6 resulted in a win - 6/12 means a 50% chance of winning if you didn't change your choice.

Anyone want to refute this?

[Skii]

So basically on paper you are increaing the probability of picking the right one by switching, whereas in reality you might as well stick with your original choice..

?

[Paul Adams]

It's a lovely problem eh?

Try this for a pseudo proof - it has a tree diagram showing the possible combinations, which Rythmic lists above, but then adds weighting to the possibility branches to show you how it's skewed:
http://www.cut-the-knot.org/peter.shtml

Another site with a very wordy explanation:
http://www.fact-index.com/m/mo/monty_hall_problem.html

Try it!
You'll "prove" to yourself that you're better off switching.

[Rythmic]

It's doing my head in now. I can't see any obvious flaws with the proofs above - but I don't believe them.

The only thing I'd say, is that it's probably invalid to count the choice of the host in, since it has no bearing on the players chance.

I'm gonna quickly go and write a vbs script (cos I can post that as plan text here, and everyone can run it without a compiler) to get empirical evidence.

[Zathras]

OK - I spent some time in excel. Heres a table containing all eventualities - if I've missed one, please let me know:
...
Anyone want to refute this?

You've forgotten the possibilities at each stage. Each event does not have equal probability of occurring.

[Blub2k]

The only thing I'd say, is that it's probably invalid to count the choice of the host in, since it has no bearing on the players chance.

but the players choice influences what he will pick.

[directhex]

don't forget that you can easily mathematically prove that 1=2

bad maths may be fun to confuse people, but don't fool yourself into believing it

[Zathras]

The host definitely picks a wrong door, so the host's selection definitely has an effect on the player's chances. Use the example posted before, 1000 doors. The player picks one, the host removes 998 wrong doors, leaving two doors, one of which definitely contains a prize. Your odds of being right on your first pick are 1000-1. Now there are two doors. Have your odds suddenly dropped to 2-1? No, because the host CANNOT remove the door you have chosen.

[Zathras]

don't forget that you can easily mathematically prove that 1=2

Err, no you can't. Your proof will be flawed, so won't be a mathematical proof at all.

[Rythmic]

Warning! Long Post!

As promised, heres the script, written in vbs so everyone can see it and run it for themselves:

Dim nLoop, nWinOnChange, nLoseOnChange, nWinOnStay, nLoseOnStay, nTries

Dim nWinningDoor, nPlayersFirstChoice, nHostsChoice, nPlayersSecondChoice

Randomize
nWinOnChange = 0
nLoseOnChange = 0
nWinOnStay = 0
nLoseOnStay = 0

nTries = 100000

For nLoop=1 To nTries
	' winning door and players first choice are random
	nWinningDoor = Int(rnd()*3+1)
	nPlayersFirstChoice = Int(rnd()*3+1)
	
	' make hosts choice
	If nWinningDoor <> nPlayersFirstChoice Then
		nHostsChoice = 6 - nPlayersFirstChoice - nWinningDoor
	Else
		If rnd()>.5 Then
			nHostsChoice = 4 - nPlayersFirstChoice
		Else
			nHostsChoice = (nPlayersFirstChoice mod 3) +1
		End If
	End If
	
	' make players second choice - can't be the hosts choice
	do
			nPlayersSecondChoice = Int(rnd()*3+1)
	loop Until nPlayersSecondChoice <> nHostsChoice
	
	' print choices
	WScript.Echo "Try: " & nLoop & " : " & nWinningDoor & "," & nPlayersFirstChoice & "," & nHostsChoice & "," & nPlayersSecondChoice

	' print & record result
	If nPlayersSecondChoice = nWinningDoor Then
		If nPlayersSecondChoice = nPlayersFirstChoice Then
			WScript.Echo "No Change, Won"
			nWinOnStay = nWinOnStay + 1
		Else
			WScript.Echo "Changed, Won"
			nWinOnChange = nWinOnChange + 1
		End If
	Else
		If nPlayersSecondChoice = nPlayersFirstChoice Then
			WScript.Echo "No Change, Lost"
			nLoseOnStay = nLoseOnStay + 1
		Else
			WScript.Echo "Changed, Lost"
			nLoseOnChange = nLoseOnChange + 1
		End If
	End If
	
	WScript.Echo vbcrlf
Next

WScript.Echo vbcrlf & vbcrlf & "Final Figures"

WScript.Echo "No Change, Won :" & nWinOnStay
WScript.Echo "Changed, Won :" & nWinOnChange
WScript.Echo "No Change, Lost :" & nLoseOnStay
WScript.Echo "Changed, Lost :" & nLoseOnChange
	
WScript.Echo vbcrlf & vbcrlf & "Percentages"

WScript.Echo "No Change, Won :" &  nWinOnStay / nTries *100
WScript.Echo "Changed, Won :" & nWinOnChange / nTries*100
WScript.Echo "No Change, Lost :" & nLoseOnStay / nTries*100
WScript.Echo "Changed, Lost :" &  nLoseOnChange / nTries*100

You can change the number of "trys" by changing the line "nTries = 100000" near the top - higher the figure, the more accurate, but slower.

Pop it into a text file - call it something like monty.vbs - Then go to the command line and run it with "cscript monty.vbs"

make sure you're in the right directory - don't just double click on it or you'll be greated with 10000 popups Seriously - don't.

My figures (from 100000 tries):

No Change, Won :13.77%
Changed, Won :33.218%
No Change, Lost :33.357%
Changed, Lost :19.655%



Thus I'm astounded - I was absolutely wrong! - you do get a better chance by changing your mind!

[Paul Adams]

Thus I'm astounded - I was absolutely wrong! - you do get a better chance by changing your mind!

Now you have experienced what I did - from being told in the pub "no, this is true, honestly" and not believeing a word of it, to reading up on various websites.
It was a jaw-dropping revelation.

Fun, eh?!

You can now see why I did Further Mathematics and aced Pure Maths and Mechanics (with a little more difficulty), but could not get my head (and tongue!) around Statistics.

Boolean algebra at uni was a little help, though once events were no longer discrete or mutually exclusive they became awkward too - I had to draw Venn diagrams to understand the questions most of the time!

[Rythmic]

Believe it or not, I aced Maths and Further Maths at A Level, and algebra at uni.
My algebra lecturer hated venn diagrams - called them "Mickey Mouse" diagrams (OK - hands up anyone who gets that joke...)

It was advanced calculus that got me - integrating is seems mostly guess work when your doing equations like dx^yn/dyx^-1 = dy^n + dx^n/dx^(n-y)

That was when I changed my degree to pure computer science, not compsci with maths.....

You can see my need to have to check everything myself though - think that generally works well with computers

[Rythmic]

You've forgotten the possibilities at each stage. Each event does not have equal probability of occurring.

Yup - I did (slaps head)

Didn't want to believe