Comments on: Deal Or No Deal vs Monty Hall

By: Jose

Jose — Sun, 02 Nov 2008 06:51:32 +0000

What if in the MH situation, you were given an opportunity of choosing up one of 10 cases. Then Monty eliminated the other choices for you one at a time, knowing that he only eliminates the wrong case. Between each elimination, you are given the option of changing cases.
He keeps doing this until there are only 2 cases left, should you switch?

By: Tim

Tim — Fri, 31 Oct 2008 21:36:51 +0000

Chris I will very simply show you why that logic is wrong and swapping makes no difference.

You are left with two cases, one wins $1m and the other $1. Now apply all the logic you and some other have done but every time you use the words “winner” or “$1m box” exchange that with “$1 box” and you will see the error. The EXACT same logic can be applied for the path of the 1$ box. If you are in the land of backwards where $1 was actually more money than $1m then according to your logic then you should swap boxes in order to win the $1 case. They can’t both be right hence the paradox.

Your 2nd paragraph is also wrong. The host knowing where the winning box is is precisely what makes it different.
If you have 3 boxes and only 1 wins and you pick a random box, the host will remove one of the losing boxes from play but he will never remove the winning box from play and that is where you gain the equity in swapping.

By: Chris Smith

Chris Smith — Fri, 31 Oct 2008 18:38:48 +0000

You can’t change your probability of choosing the winning box based solely on the knowledge you have obtained through the game, as your sample set doesn’t change as you go, you are just seeing it. From the beginning, if there are 26 boxes from which to choose and only one winner, you have a 1/26 chance of winning. So if you are so fortunate as to have seen all of the others but one, you need to swap boxes. Even when you get to the end, you are still looking at the same initial conditions and the same sample set (the money didn’t change boxes during the game), but it just happens that you have seen 24 other boxes.

This idea of a non-changing sample set is why the Monty Hall problem works out as it does. It has nothing to do with the host or the fact that the host knows where the winning box is. If Howie and “The Banker” both know where the winning box is, as long as Howie can bluff it and “The Banker” makes his side offers in a standard way, the game is still “fair.”

By: Dalton

Dalton — Sun, 19 Oct 2008 08:05:39 +0000

Suppose you pick your case, and then you pick the last remaining case. Both of these have the same probability of being the top prize (1/26; US version). Do you agree?

Yes, okay, then, you go on and open 24 cases. If you’re lucky to never reveal the top prize, what are the probabilities of the two remaining cases? You’ve already agreed they’re the same.

What makes this different from MH is that in MH you always reach the opportunity to swap. In DoND, it’s highly unlikely to get to two final cases and still have the top prize left.

This could also be simulated with a Python program; you’ll have to throw out a bunch of scenarios until you get to a scenario where one of the last two has the top prize. About 500 times out of 1000, you will have picked the top prize initially.

import random

iters = 1000
random.seed()
million = 1000000

count = 0
picked = 0

while (count < iters):
cases = [0.01,1,5,10,25,50,75,100,200,300,400,500,750,\
1000,5000,10000,25000,75000,100000,200000,300000,\
400000,500000,750000,million]
random.shuffle(cases)

yours = random.choice(cases)
del cases[cases.index(yours)]

for i in range(len(cases)-1):
pick = random.choice(cases)
del cases[cases.index(pick)]

if yours == million or million in cases:
count = count + 1
if yours == million:
picked = picked + 1

print “out of %s times, your initial pick was top %s times” % (count,picked)

By: Ben Stilwell

Ben Stilwell — Mon, 15 Sep 2008 16:57:53 +0000

Its a problem I began thinking about last night after watching the movie ‘21′ which included the MH problem.

In the end I wasn’t sure, on the one hand the losing doors are opened on purpose and on the other they are opened randomly and luckily. Does this change probability? I basically went back and forth with my reasoning thinking it could be 50/50 or maybe 25/26(US version of the game).

I had settled on that it was advantageous to switch. When you pick a case you have 1/26 chance of hitting the million and the odds of not selecting the million being 25/26. Which means that the odds of the million being in what of the ladies cases is 25/26. As cases are eliminated these odds shouldn’t change.

I do however understand Johns last point about the Goat having similar odds, so by tomorrow I may change my mind :p. But if you look at it as 1 car vs 25 goats, does the goat really have similar odds?

By: John Langley

John Langley — Thu, 28 Aug 2008 20:39:12 +0000

Thank you James Casbon. Now I’m the idiot because I thought I knew what I was talking about. I apologize for my abrasive language.

I’m back to believeing it’s a 50/50 split when we get down to the final two cases. In my scenario I assumed the $1 million took the path down to the final two cases. But, just as easily, the goat could have taken that path too. In the end, I am left with two cases that took a path. I don’t know which scenario happened.

Monty Hall knows which scenario happens. He eliminated the goat for you. So, in 21 of the examples, you are going to be presented with an ending scenario that involves $1 million and a goat. But, you really don’t desirve this because on DNOD, you would have arleady eliminated the $1 million earlier. You woudl never get to a Monty Hall scenario at the end. The only time you do get to the end of having $1 million in a case and a goat int he other case is the 50/50 case.

I’m not an Einstein. But, Einstein proved all of his theories through logic and reasoning. I don’t like numbers because I work with them as part of my job. I know how they can get manipulated. I value companies. OMG. You just wouldn’t believe the things that people do with numbers when it comes to valuing a company….

If I could delete my comment sent yesterday, I would. Not becuase I was wrong. But, because I called all of you nice people idiots.

By: James Casbon

James Casbon — Tue, 26 Aug 2008 10:58:30 +0000

Thanks, John for your analysis, and for the show Cops

That ‘pretentious’ ’stupid probability math’ is backed up by a simulation that requires no probability maths at all. Have you found a flaw in that as well?

By: John Langley

John Langley — Tue, 26 Aug 2008 08:24:41 +0000

Hey idiots. Swap the case. Monty Hall DOES apply because you are given knowledge throughout the game - just as if Monty Hall was telling you a goat was inside of each case eliminated. It doesn’t matter if Monty Hall tells you this, or if you just get lucky on DOND. The fact remains that after your choice is made, you are given knowledge that $1 million still remains. This is new knowledge. By the time you get down to the end with only two cases remaining, you KNOW that one of the cases has $1 million, and the other case does not. This knowledge could have been given to you by Monty Hall, Helen Keller, Jack Benny, or by your stupid luck. Nonetheless, you are left with two cases and one has $1 million. You know that one of them has $1 million. If the total number of cases is 22, then the odds that your case has $1 million is 1 in 22 (these are the original odds when you picked your case). The odds that the remaining case has $1 million is 21 in 22.
Switch cases. The Monty Hall principle does apply.
It is a completely different story if Howie did not tell you that you were left with two cases, one with $1 million and the other with something else. Pretend that you started with 22 cases and just randomly eliminated 20 cases. The big information board isn’t working, and hot chicks holding the cases never open the cases. They just sit there looking stupid and so damn hot, and they forget to open the cases to show you what you just eliminated. So, you have just eliminated 20 cases. You don’t know what you eliminated. Your thumb is up your ass and you are now left with two cases. Should you switch? You should switch only if it gives you the opportunity to walk up to the hot model holding the case. It does nothing for your odds of winning. But, hopefully you were able to get a little sniff of her perfume. Odds are you already lost the $1 million with one of your earlier eliminations anyway. You don’t know anything about the last two remaining cases. If you switch or not, the odds are still 1 in 22 that you picked $1 million.

Take that stupid probability math and stick it where the sun doesn’t shine. It’s so pretentious. Dont’ let someone confuse you with flawed logic even if they use fancy numbers.

By: Andrew

Andrew — Fri, 27 Jun 2008 12:59:41 +0000

Tom, you really haven’t thought that through. The probability that the OTHER box started with a big one is ALSO 5/26. Your own (flawed) logic could equally be applied from the perspective of the other box, to say you shouldn’t swap.

The odds do NOT remain the same as the start because every box that is opened change the odds of what is in your box and every other box. According to your logic, you would still have a 1/26 chance of having the sum that is in a box that has already been opened. Clearly, once a box has been opened, the probability that you chose that box at the start is no longer in existence, hence the probabilities continue to evolve rather than remaining consistent with the initial probabilities.

By: Tom

Tom — Fri, 27 Jun 2008 05:37:03 +0000

I was just watching DOD and the scenario unfolded where the guy had three cases: $100, $10,000 & $750,000.

He took the deal, but when asked what he would done he said Case 3 ($10,000). He then said he would swap (which I thought he should of bc of the MH problem). Had he swapped he would have won the $750k.

I was trying to explain why he should have swapped to my roommate, but I’m not much smart in the maths, so came to the Internet. I was interested to see such debate and to see the consensus is a no swap.

I disagree and say they should swap.

Think of it this way:

There are 26 cases. You obv. have 1-26 of picking the $1M. But let’s not limit it just to that. Really, there are about 5 Big Money cases. So let’s say those are the cases you are after. The rest are Crap Cases

Your odds are 5-26 or picking a Big Money Case, and 21-26 of picking a Crap Case.

So from the start, the odds are far likelier that you picked a Crap Case vs. a Big Money Case.

If you have gotten down to 2 cases, one we know is a Crap Case and the other a Big Money Case, it doesn’t matter if it was revealed to you intentionally or by random, you are still at that same point. It is irrelevant how you got there.

I know the perception is it is 50/50 between the Crap vs. Big $ case because you see the two in front of you, but I refuse to believe it.

When you picked the cases there was a 21/26 chance you picked a Crap Case. Thus, when given the option, you should swap your Crap Case with what is likely to be a Big $ Case. You have now inversed your odds of being 5/26 that you now have a Crap Case. i.e. 21/26 of having a Big $ Case.

Again, I am not a statistician by any means, but this sounds pretty logical to me, and in spirit with the MH problem, which everyone seems to think doesn’t hold water in this case. (No pun intended.)