# Randomness, coin flipping and probability **URL:** \ **Category:** Factual Questions\ **Created:** [February 20, 2003, 12:14am UTC](https://boards.straightdope.com/t/randomness-coin-flipping-and-probability/156236 "2003-02-20T00:14:45Z")\ **Posts on this page:** 14\ **Page:** 3
**Author:** ![DarrenS](https://avatars.discourse-cdn.com/v4/letter/d/977dab/32.png) [@DarrenS](https://boards.straightdope.com/u/DarrenS)\ **Post date:** [February 20, 2003, 8:17pm UTC](https://boards.straightdope.com/t/randomness-coin-flipping-and-probability/156236/41 "2003-02-20T20:17:08Z")
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_ ? ---
**Author:** ![CurtC](https://avatars.discourse-cdn.com/v4/letter/c/ce73a5/32.png) [@CurtC](https://boards.straightdope.com/u/CurtC)\ **Post date:** [February 20, 2003, 9:34pm UTC](https://boards.straightdope.com/t/randomness-coin-flipping-and-probability/156236/42 "2003-02-20T21:34:36Z")
> [@](#): > > _davejg wrote:_ > One thing Monty would do from time to time was allow you to change your mind. Contestant initially picks curtain 1, Monty says, “well, let me show you what’s behind curtain 2”, and it’s a boobie prize." 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. > [@](#): > > Statistically, you should always change your mind. In the above situation, chances are 1 in 3 that curtain #1 is correct. Chances are 50/50 that it’s behind curtain 3. 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. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [February 20, 2003, 11:42pm UTC](https://boards.straightdope.com/t/randomness-coin-flipping-and-probability/156236/43 "2003-02-20T23:42:02Z")
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. ---
**Author:** ![davejg](https://avatars.discourse-cdn.com/v4/letter/d/ec9cab/32.png) [@davejg](https://boards.straightdope.com/u/davejg)\ **Post date:** [February 21, 2003, 6:28am UTC](https://boards.straightdope.com/t/randomness-coin-flipping-and-probability/156236/44 "2003-02-21T06:28:39Z")
\>\> 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. ---
**Author:** ![CurtC](https://avatars.discourse-cdn.com/v4/letter/c/ce73a5/32.png) [@CurtC](https://boards.straightdope.com/u/CurtC)\ **Post date:** [February 21, 2003, 8:56pm UTC](https://boards.straightdope.com/t/randomness-coin-flipping-and-probability/156236/45 "2003-02-21T20:56:13Z")
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. ---
**Author:** ![Manduck](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/manduck/32/256_2.png) [@Manduck](https://boards.straightdope.com/u/Manduck)\ **Post date:** [February 22, 2003, 1:39am UTC](https://boards.straightdope.com/t/randomness-coin-flipping-and-probability/156236/46 "2003-02-22T01:39:39Z")
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. ---
**Author:** ![gazpacho](https://avatars.discourse-cdn.com/v4/letter/g/6f9a4e/32.png) [@gazpacho](https://boards.straightdope.com/u/gazpacho)\ **Post date:** [February 22, 2003, 1:59am UTC](https://boards.straightdope.com/t/randomness-coin-flipping-and-probability/156236/47 "2003-02-22T01:59:26Z")
> [@](#): > > \*Originally posted by Manduck \* > \*\*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. ---
**Author:** ![Punoqllads](https://avatars.discourse-cdn.com/v4/letter/p/d2c977/32.png) [@Punoqllads](https://boards.straightdope.com/u/Punoqllads)\ **Post date:** [February 22, 2003, 2:36am UTC](https://boards.straightdope.com/t/randomness-coin-flipping-and-probability/156236/48 "2003-02-22T02:36:00Z")
> [@](#): > > \*Originally posted by DarrenS \* > \*\*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_ ? \*\* You mean [http://www.straightdope.com/classics/a3\_189.html](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. ---
**Author:** ![Manduck](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/manduck/32/256_2.png) [@Manduck](https://boards.straightdope.com/u/Manduck)\ **Post date:** [February 22, 2003, 3:48am UTC](https://boards.straightdope.com/t/randomness-coin-flipping-and-probability/156236/49 "2003-02-22T03:48:27Z")
> [@](#): > > \*Originally posted by gazpacho \* > \*\*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. \*\* 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: ---
**Author:** ![Richard\_Pearse](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/richard_pearse/32/13144_2.png) [@Richard\_Pearse](https://boards.straightdope.com/u/Richard_Pearse)\ **Post date:** [February 22, 2003, 5:04am UTC](https://boards.straightdope.com/t/randomness-coin-flipping-and-probability/156236/50 "2003-02-22T05:04:22Z")
> [@](#): > > \*Originally posted by Manduck \* > \*\*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. ---
**Author:** ![CurtC](https://avatars.discourse-cdn.com/v4/letter/c/ce73a5/32.png) [@CurtC](https://boards.straightdope.com/u/CurtC)\ **Post date:** [February 22, 2003, 6:45am UTC](https://boards.straightdope.com/t/randomness-coin-flipping-and-probability/156236/51 "2003-02-22T06:45:04Z")
> [@](#): > > _Manduck wrote:_ > It doesn’t matter what his motivation is, because it is given that he always reveals an empty door. 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.” ---
**Author:** ![MonkeyMensch](https://avatars.discourse-cdn.com/v4/letter/m/82dd89/32.png) [@MonkeyMensch](https://boards.straightdope.com/u/MonkeyMensch)\ **Post date:** [February 22, 2003, 6:55am UTC](https://boards.straightdope.com/t/randomness-coin-flipping-and-probability/156236/52 "2003-02-22T06:55:20Z")
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. ---
**Author:** ![Manduck](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/manduck/32/256_2.png) [@Manduck](https://boards.straightdope.com/u/Manduck)\ **Post date:** [February 22, 2003, 7:14am UTC](https://boards.straightdope.com/t/randomness-coin-flipping-and-probability/156236/53 "2003-02-22T07:14:42Z")
> [@](#): > > \*Originally posted by CurtC \* > \*\*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.” \*\* Note the last line of davejg’s post, which is: > [@](#): > > NOTE: The KEY here is that Monty _always_ shows you a boobie prize curtain before allowing you to change your mind. 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. ---
**Author:** ![Panopticon](https://avatars.discourse-cdn.com/v4/letter/p/d2c977/32.png) [@Panopticon](https://boards.straightdope.com/u/Panopticon)\ **Post date:** [February 22, 2003, 12:22pm UTC](https://boards.straightdope.com/t/randomness-coin-flipping-and-probability/156236/54 "2003-02-22T12:22:49Z")
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 [Previous page](https://boards.straightdope.com/t/randomness-coin-flipping-and-probability/156236.md?page=2)