I seem to remember a massive discussion on the Monty Hall problem, which involved Cecil and others…and wasn’t this one of the only times that Cecil turned out to be (gasp) wrong ?
There’s a problem right here with your problem statement. In your example, there’s not enough information. Monty does this only “from time to time.” What does he use to choose when he does allow the choice and when he doesn’t? The problem works with the 1/3 - 2/3 split only when it’s stated correctly. You have to explicitly state that Monty is forced to show you a non-winning door every time.
To highlight why this is true, I have an example I like to use. Let’s say you come upon a street hustler who offers you to play the shell game. He shuffles them around, you make your pick, and then, without showing you your pick, he picks up an empty shell and asks whether you want to switch. Should you? Only a fool would say yes.
So what happens the remaining 1/6 of the time? Actually, this is the exact same thing that Click and Clack said as the solution to their weekly puzzler way back when, before even Cecil or Marilyn vos Savant addressed it. So you’re in good company.
My personal favorite way to view the law of averages: Suppose that I decide I’m going to flip a perfectly fair coin a million times. What’s the most likely number of heads? 500,000. I wouldn’t be surprised to get 499,999 heads, but it’s slightly more likely that I’ll get 500,000.
So, I flip my coin the first ten times, and lo and behold, I get ten heads in a row! I’ve now got 999,990 flips left in my experiment. Out of those remaining flips, what’s the most likely number of heads? Half of them, or 499,995. So at this point, out of all million flips, the most likely number of heads is 500,005.
Now, when I had ten more heads than tails, and I only had ten flips, wow, that was significant. But when heads outnumber tails by ten out of a million flips, that’s boring. The Law of Averages does not work by correcting past imbalances; it works by swamping out past imbalances and making them insignificant.
>> One thing Monty would do from time to time was allow you to change your mind.
On the show, Monty didn’t always show you what was behind a different door. Often he would make people stick with their initial choice.
That has nothing to do with the probablity puzzler however. I should have been more explicit.
Actually, it has everything to do with the puzzler. The answer depends on Monty’s motivations that he uses to decide whether to offer the choice. If you don’t know his motivations, you have to guess, and a reasonable guess could lead to a 50/50 answer.
It doesn’t matter what his motivation is, because it is given that he always reveals an empty door. That constrains his actions so that he can’t be acting strategically. He is just a mechanical part of the puzzle. But this has been thrashed out exhaustively in numerous old thread; you should just search up one or two of them rather than rehashing the Monty Hall puzzle yet again on these boards.
If he only offered you the choice based on if the door you chose had the good prize to try and trick you into taking the other door you would be foolish to switch.
You mean http://www.straightdope.com/classics/a3_189.html? In Cecil’s defense, the original question was worded poorly. As others have noted, he was right or wrong depending upon the constraints you assume Monty would follow.
He always offers the choice. That’s part of the formulation of the puzzle. I can’t believe I’m being sucked into another Monty Hall thread. Especially when it’s a hijack. I may never know if *DarrenS found the pie explanation helpful :mad:
It has been stated above that he only sometimes reveals a no prize door and that when he does this, he always offers the choice to change. The question is over his motivation in revealing a no-prize door. If he does this because he knows that you’ve picked a prize door then it takes all the statistics away and you would be foolish to switch. If on the other hand, he does it randomly, then statistics seem to indicate you should switch.
I refer you back to davejg’s post earlier in this thread. He is the one who brought the Monte Hall puzzle in here, and he said “One thing Monty would do from time to time was allow you to change your mind.”
So in this case it was not a given that he always reveals an empty door. In fact, when I’ve heard people state the puzzle, they hardly ever state that this is a given. Davejg went as far to state that he does it only “from time to time.”
I followed, years ago, the whole Monty, vos Savant, SciAm, and of course Cece fun re this question.
I just thought that I would add one thing. If Monty is being malicious, say at the behest of his budget strapped producers, then there is NO reason to ever change ones choice.
It avoids descending into a regress of if he knows, that I know, that he knows that I know, that he knows…
The upshot is Malicious Monty can’t drive your probability below .33 with this strategy. FWIW.
Note the last line of davejg’s post, which is:
When he said “from time to time”, he was talking about the actual TV show; in the statistical puzzle Monty always reveals a door, as stated by davejg at the end of his post.
I just wanted to thank everyone posting on this thread for breakign this stuff down so well. We’ve a fascinating and valuable community here.
Cheers,
panopticon