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Perplex City Puzzle Cards
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PXC: Purple Puzzle Cards
#174 - Purple -Jaunty Paul
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[quote="hobyrne"]Okay, since this _is_ a confusing issue, here is a worked-out example. The three prizes are: Goat 1, Goat 2, Yacht. Making a distinction between the goats is important. If you flip two coins, there are three outcomes: two heads, one heads and one tails, two tails. If you flip a nickel and a dime, there are four outcomes: nickel heads & dime heads, nickel heads & dime tails, nikel tails & dime heads, nickel tails & dime tails. Monty Hall has the following instructions: If the person chooses Goat 1, let him have it. If the persion chooses Goat 2, open the door with Goat 1. If the person chooses the yacht, open the door with Goat 1. Chooses goat 1: Happens 1/3 of the time. Contestant loses. Chooses goat 2: Happens 1/3 of the time. Contestant gets choice: wins if he switches. Chooses yacht: Happens 1/3 of the time. Contestant gets choice: wins if he doesn't switch. So, the contestant gets a choice 2/3 of the time. HALF of those times he gets a choice, he should switch, HALF of the times he gets a choice, he shouldn't switch. BTW: Interesting end note. If the contestants switch half the time, the overall chance of winning the prize is 1/3... exactly the same as if this little game were done away with and he just always got his first choice. The extra game is just for show, to make it *seem* more exciting for TV.[/quote]
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hobyrne
Okay, since this _is_ a confusing issue, here is a worked-out example.
The three prizes are: Goat 1, Goat 2, Yacht. Making a distinction between the goats is important. If you flip two coins, there are three outcomes: two heads, one heads and one tails, two tails. If you flip a nickel and a dime, there are four outcomes: nickel heads & dime heads, nickel heads & dime tails, nikel tails & dime heads, nickel tails & dime tails.
Monty Hall has the following instructions: If the person chooses Goat 1, let him have it. If the persion chooses Goat 2, open the door with Goat 1. If the person chooses the yacht, open the door with Goat 1.
Chooses goat 1: Happens 1/3 of the time. Contestant loses.
Chooses goat 2: Happens 1/3 of the time. Contestant gets choice: wins if he switches.
Chooses yacht: Happens 1/3 of the time. Contestant gets choice: wins if he doesn't switch.
So, the contestant gets a choice 2/3 of the time. HALF of those times he gets a choice, he should switch, HALF of the times he gets a choice, he shouldn't switch.
BTW: Interesting end note. If the contestants switch half the time, the overall chance of winning the prize is 1/3... exactly the same as if this little game were done away with and he just always got his first choice. The extra game is just for show, to make it *seem* more exciting for TV.
Posted: Tue Oct 03, 2006 8:37 am
hobyrne
Precisely what FairmontKing says. It depends on Jaunty's motivations and instructions. If the director of the show tells him, "You *must* open a goat door", that is one scenario. However, in the eerily coincidental [/sarcasm] parallel case of Monty Hall, Monty did _not_ *have* to open a door. It made for better TV, when he did, but he did not have to every time.
Therefore, if Monty was stupid, and wanted to save the network money, then every time someone chose a goat, he would end the game at that decision and the contestant is a loser, and every time someone chose the yacht, he would continue the game and give the contestant another chance to lose.
Of course, in this case, people would soon realise that the only time they are ever given a choice is when they have the winning door to begin with. So, what Monty really did, was to sometimes open a goat door when the contestant chose a goat. So, the best Monty could do to save the network money, is: If the contestant chooses the yacht, open a goat door. If the contestant chooses a goat, open the other goat door 1/2 of the time.
Conversely, if Monty wanted the contestants to win, he could end the game whenever someone chooses right, and give the contestant another chance to win if he chose wrong.
The card does not say that Jaunty *had* to open a goat door, just that he _did_. For that reason, the probabilities are 0.5 and 0.5.
Posted: Tue Oct 03, 2006 8:20 am
FairmontKing
I studied this problem as an example in a graduate-level decision analysis class. I, too, struggled with the given solution at first. The reason the answer is not as straightforward as it should be is because we really don't know Jaunty's intentions and motives.
For the given answer to be correct, the following assumptions must be true:
1. Jaunty knows which door the prize is behind.
2. Jaunty decided ahead of time that regardless of which door you chose, he was going to show you a door without the prize.
Basically, our odds only improve if we switch because Jaunty is trying to help us.
Posted: Sat Sep 30, 2006 6:55 pm
X9Tim
GreenWindmill wrote:
Look at it this way:
There are therefore 3 possible scenarios:
You pick door 1: The host eliminates door 2, you're left with 1 & 3. If you switch you win the yacht.
You pick door 2: The host eliminates door 1, you're left with 2 & 3. If you switch you win the yacht.
You pick door 3: The host eliminates door 1, you're left with 2 & 3. If you don't switch you win the yacht.
Hhmmm, what about scenario 4:
You pick door 3: The host eliminates door 2, you're left with 1 & 3. If you don't switch you win the yacht.
Then there are 4 scenarios and you win in 2 of them whether you switch or not.
I've read this type of problem and its explanation once or twice before, and I'm still not sure it's really right....
I DID post 1/2, 1/2 first time without being sure, but I knew what the other "correct" solution was....
Still I've got the points and thats what counts
Posted: Fri Sep 29, 2006 10:49 am
GreenWindmill
Quote:
No, no, it was useful.
Woo hoo! A useful post from the Windmill - that's a definite first!
Posted: Tue Sep 19, 2006 4:48 am
Factor41
Aaaahhhh
GreenWindmill wrote:
Not that this post is any real use...
No, no, it was useful. It's all in the wording. It is the probability that you will get the prize if you switch, not the probability that you will pick the correct one of two doors. Makes more sense if you say it like that, and with those scenarios spelt out.
Thanks.
Posted: Mon Sep 18, 2006 4:53 pm
GreenWindmill
Look at it this way:
Let's assume the yacht is behind door 3 each time.
The gameshow host is always going to leave the door you pick and one other (which will be door 3 if you haven't picked it, and for argument's sake, door 1 if you did pick 3).
There are therefore 3 possible scenarios:
You pick door 1: The host eliminates door 2, you're left with 1 & 3. If you switch you win the yacht.
You pick door 2: The host eliminates door 1, you're left with 2 & 3. If you switch you win the yacht.
You pick door 3: The host eliminates door 1, you're left with 2 & 3. If you don't switch you win the yacht.
You win the yacht by switching in 2 of the 3 scenarios. In other words there's a 2/3 chance of winning if you switch and 1/3 if you don't.
As people have stated previously, this is more obvious with more doors - consider it with 100.
Not that this post is any real use, the correct answer has been confirmed looong ago, just trying to convert some non-believers!
Posted: Wed Jul 05, 2006 6:02 am
Ashin
Re: Ringsting:
mama_needs_a_new_car wrote:
I absolutely agree - the frank fact is that we end the contest with two doors - and either a goat or a yacht behind each. 50/50. Two choices, two doors, two prizes - equal probability.
The other numbers (1/3, 2/3) are only in play within the context of the whole contest.
Hey dude. You're right that the probability of winning and and losing is equal (50/50). That's not what the question is asking, the question is "What is your probability of winning if you switch?"
The 50% winning field gets split into thirds, 2 of which are switching, and 1 of which is not-switching.
Start liking it.
Posted: Thu Jun 29, 2006 4:24 pm
oliverkeers13
No, similar to the theory of evolution, in that way. No matter how much sense it makes, some people will insist on believing in creationism.
Posted: Thu Jun 22, 2006 3:17 am
mama_needs_a_new_car
Ringsting:
I absolutely agree - the frank fact is that we end the contest with two doors - and either a goat or a yacht behind each. 50/50. Two choices, two doors, two prizes - equal probability.
The other numbers (1/3, 2/3) are only in play within the context of the whole contest.
I will assume the host was always going to eliminate one of the three doors and leave the contestant with two choices - it is like forcing a card in a magic trick.
I will bow down to the mathmatic "authorities" and submit the answer they want (like taking a test in school with a lousey teacher who just wants you to regurgtate his/her own opinions), but I don't have to like it.
Posted: Thu Jun 22, 2006 12:00 am
ringsting
I think that where the card isn't clear is that it dosn't explain at what point in time the probability should be taken.
at point x in time, before any of the doors have been opened; the probability that A is correct is 1/3 and the probability that its not correct is 2/3. Hence when C is opened B becomes 2/3 as C=0 (according to the original calculation.)
however we move on in time and at point y in time, after C has been opened; we are simply left with a fresh choice of 2 doors, giving 50/50.
I think that you have to start each situation with a new calculation, otherwise you have to take into account all preceeding events such as: was the contestant chosen from a group of 100 people? if so you have the odds of 1/100 added onto the calculation ect
Posted: Wed Jun 21, 2006 6:38 pm
Fuseunderground
forensick wrote:
I think the biggest problem with this question is that it's
There is no problem with the question,
It's just about your choices, not about 'doors'.
This card (like 'Illogical' and others) shows that the way people think about a problem,
is not necessarily the right way (In maths/logic/or whatever)
The correct answer for this puzzle reflects the situation in real life, honestly.
Posted: Fri Apr 28, 2006 1:23 pm
forensick
I think the biggest problem with this question is that it's
a) not made clear or
b) we just don't put it together
that the host is always going to show the other goat first.
He's not going to walk over and open the door with the car first (if you chose a goat). It's that logic that made the problem difficult for me.
Don't get me wrong, I'm still not a fan of the math behind it. I'd much rather accept 50/50 or even 66/50 before the "truth."
Posted: Fri Apr 28, 2006 12:19 pm
Dragon_Hogg
solved
the exepted answer is
Spoiler (Rollover to View):
Probability for Door A: 1/3
Probability for Door B: 2/3
but the propre answer is 50:50 or 1/2
cos you only get a choice of 2 doors
as 1 is always taken away so 1 door+goat
is taken away you only chose between the two left.
so (50+50+50)-50=100, stupid mathematicians.
Posted: Wed Mar 08, 2006 4:13 pm
wizzsteve2
can i get an answer to this plz ive tried 2 answers
Spoiler (Rollover to View):
a=1/3b=2/3 and a&b=2/3
and both failled so can i get an answer that works plz
Posted: Sun Mar 05, 2006 6:16 am
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