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Topic: You are on a game show (Read 1623 times) |
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UgoLocal02
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You are on a game show
« on: Jun 12th, 2014, 2:18am » |
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You are on a game show. You are shown three closed doors. A prize is hidden behind one, and the game show host knows where it is. You are asked to select a door. You do. Before you open it, the host opens one of the other doors, showing that it is empty, then asks you if you'd like to change your guess. Should you, should you not, or doesn't it matter?
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dudiobugtron
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Re: You are on a game show
« Reply #1 on: Jun 12th, 2014, 1:02pm » |
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I'll keep my door, Monty. I know everyone says you should switch, but I can smell a goat a mile away.
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rloginunix
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Re: You are on a game show
« Reply #2 on: Jun 12th, 2014, 7:50pm » |
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Small nitpick: the last four words of the problem statement should likely read "or does it matter".
That's the question asked by the actor Kevin Spacey in the movie "21" where he plays an MIT math professor.
With some assumptions that transform the real world into the math world I think that the numbers favor the switch.
Assumptions. I don't watch much TV, let alone these shows, but I would guess the host 1) knows very well where the prize is and (semi hence) 2) will never pick the prized door.
"No switch" strategy. The chances of you winning the prize are 1/3. Whether the host reveals the no-prize door or not does not really change anything. (S)he opens the door, nothing there, you stick to your guns, still 1/3.
"Switch" strategy. I think that the trick here is that with this strategy the moment you decide to switch you reset the case count - there are 3 cases to consider: the prize is behind the door 1, door 2 or door 3.
Analysis.
Let's for argument sake assume that you picked the door 3 for some reason. So door 3 is your original choice and you will switch.
Case 1: the prize is behind the door 1. This eliminates doors 1 and 3 for the host's choices - one has the prize behind it (the host knows it and will not open that door) and the other is your choice - (s)he can only open the door 2. Since you decided to switch there's no wiggle room here: you switch from door 3 (original choice) to door 1 (the only one left closed) and ... bingo! You win a prize.
Case 2: the prize is behind the door 2. This eliminates doors 2 and 3 for the host's choices - (s)he can only open the door 1. You switch and ... bingo! You win a prize.
Case 3: the prize is behind the door 3. This eliminates only door 3 (your own choice) for the host - (s)he can flip a coin and pick either door 1 or 2, doesn't matter. You switch and ... get a one way ticket to Loserville.
Tally the chances up: it looks like that with switching your chances to win are 2/3.
I also think you can look at it from the "inversion" perspective - if you decide to switch then you lose (inverse of win) iff you pick the prized door on the first go: 1 out of 3. Which means that you you are pretty much guaranteed to win (inverse of lose) in all the other cases: 2 out of 3.
No? Where's the hole?
P.S
I think the above analysis is the talk show host behavior-sensitive: change how (s)he makes a decision which door to open and the numbers would change.
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towr
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Re: You are on a game show
« Reply #3 on: Jun 12th, 2014, 10:50pm » |
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on Jun 12th, 2014, 7:50pm, rloginunix wrote:| Small nitpick: the last four words of the problem statement should likely read "or does it matter". |
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Why? It doesn't make a semantic difference, and I'd say the sentence flows better this way.
Anyway, this puzzle is just the Monty Hall problem, which can be found all over the forum -- even with the crappy search this forum offers 
Monty Hall:
http://www.ocf.berkeley.edu/~wwu/riddles/easy.shtml#montyHallShow
http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_eas y;action=display;num=1028741912
http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_eas y;action=display;num=1165632993
http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_eas y;action=display;num=1209106543
Variants:
http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_har d;action=display;num=1168923030
http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_eas y;action=display;num=1188666359
http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_eas y;action=display;num=1209322639
http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_med ium;action=display;num=1216276908
http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_med ium;action=display;num=1325686088
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rloginunix
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Re: You are on a game show
« Reply #4 on: Jun 13th, 2014, 8:40am » |
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My feeling was that "doesn't" sounds more suggestive and affirmative in the sense of "... doesn't it matter that so and so ...". In other words it already offers a clue that yes, it does matter, now go figure out why. The "does" formulation feels a bit more neutral, to me that is. Just a nitpick, no big deal.
As far as posting the answer - my bad. I don't search the forum before posting a solution. I assume the author checks for duplicates (as I do).
With so many existing versions of the problem I doubt I added anything new.
Delete it.
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towr
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Re: You are on a game show
« Reply #5 on: Jun 13th, 2014, 12:10pm » |
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That list of duplicates wasn't really in response to your answer; but maybe that would have been clearer if I had switched the order.
Checking for duplicates is indeed something the author ought to check (there's a sticky topic on nearly every board saying as much).
On the plus side, it should now be even easier to find a duplicate
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jo
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Re: You are on a game show
« Reply #6 on: Aug 5th, 2014, 4:48am » |
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@rloginunix
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JaneBD
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Re: You are on a game show
« Reply #7 on: Aug 7th, 2014, 1:41am » |
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on Jun 12th, 2014, 1:02pm, dudiobugtron wrote:| I'll keep my door, Monty. I know everyone says you should switch, but I can smell a goat a mile away. |
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Haha, I wish this was the actual naswer!
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MathsForFun
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Re: You are on a game show
« Reply #8 on: Aug 29th, 2014, 1:28pm » |
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A twist: you receive intelligence that an important terrorist leader is hiding in one of 3 caves - cave 1, cave 2 or cave 3. You scramble a fighter aircraft armed with a missile to attack cave 1. While the fighter is on its way, you receive a further report that he's not in cave 2. Should you order the fighter to abandon the attack on cave 1, and attack cave 3 instead?
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towr
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Re: You are on a game show
« Reply #9 on: Aug 29th, 2014, 1:40pm » |
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It's unlikely there's a causal link between your initial choice and the later report; so it makes no difference.
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MathsForFun
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Re: You are on a game show
« Reply #10 on: Aug 29th, 2014, 1:57pm » |
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on Aug 29th, 2014, 1:40pm, towr wrote:| It's unlikely there's a causal link between your initial choice and the later report; so it makes no difference. |
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This is correct in terms of the probability - and clearly explained - good show! - but... other things being equal, changing a military plan during its operation will reduce its chances of success - so you should stick to the original choice. Also, given that you chose cave 1 to attack, it is likely to be a better target in some way.
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Grimbal
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Re: You are on a game show
« Reply #11 on: Sep 2nd, 2014, 1:04am » |
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It could be that the rebels do hide in cave 1 and they got word that an attack is prepared on cave 1. So they decided to "accidentally" release information showing cave 2 is empty.
Maybe someone was given the mission of checking by all means whether the rebels are in cave 2. He got the information that they are in cave 1. So he concludes they are not in cave 2 and reports that conclusion.
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Annettagiles
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Re: You are on a game show
« Reply #12 on: Sep 23rd, 2014, 4:19am » |
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No I won't change my option
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wakiza33
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Re: You are on a game show
« Reply #13 on: Sep 23rd, 2014, 11:00am » |
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The probabilities remain the same.
What does change, though, is behavioral and non-verbal cues. I would make my decision based on my internal perception of the host's and producer actions and subliminal cues.
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dudiobugtron
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Re: You are on a game show
« Reply #14 on: Sep 23rd, 2014, 1:33pm » |
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on Sep 23rd, 2014, 11:00am, wakiza33 wrote:| What does change, though, is behavioral and non-verbal cues. I would make my decision based on my internal perception of the host's and producer actions and subliminal cues. |
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Yes, but one of those 'non-verbal cues' is that the host probably wouldn't open the door that had the prize in it.
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