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On July 16 2011 07:14 hypercube wrote:Show nested quote +On July 16 2011 02:38 Chill wrote:
Random:
HH x
HT o
TH o
TT x
50%
After knowing at least one was heads:
HH x
HT o
TH o
TT
66.6%
O_o
I think you're just thinking about it wrong. His knowledge isn't affecting the outcome - the outcome was random but he gave you specific information about the result that eliminates one of the possibilities.
If he flipped a coin, told you it was heads, and then flipped the other coin, there would be a 50% chance one of them was tails. But that's not the same thing. In scenario 1, TT was an option was that later eliminated with information. In scenario 2, TT is never a possibility so it doesn't factor in. How about this: Random: HH -> H HT -> T TH -> T TT -> H 50% After knowing at least one was heads: 1 x HH -> H 0.5 x HT -> 0.5T 0.5 x TH -> 0.5T TT50% I don't see a clear reason to prefer your distribution to mine. For example if he just looked at one of the coins in secret and told what he saw then the second distribution is the correct one (because for HT and TH he'd say tails 50% of the time). edit: to clarify, everyone who got 2/3 was assuming that every time one of the coins comes up heads your friend will say so. If that's true you are correct to just count up the cases. However, this assumes that he has an a priori preference for heads, i.e for HT he'll always say one of them is heads and never that one of them is tails. I just don't feel like that assumption is justified, given the description of the problem.
I'm having a hard time figuring out why you're giving double the probability of HH occurring there. That's an assumption nobody made at all. The only assumption made is that your friend flipped two coins and you know that ONE of them came up heads.
The most important piece of information to realize in the entire puzzle is that your friend doesn't specify WHICH SPECIFIC coin came up Heads. Only that one of them did. Out of the four potential possibilities, the clue he gives you only rules out ONE of them, not two of them.
Hypothetically, if he told you that the FIRST coin came up heads, then your outcome list would look like this (remember that the first letter represents the first coin, and the second letter represents the second coin):
HH
HT
TH
TT
And you'd have a 50:50 chance, of picking the right answer. But, that's NOT what he told you. There's a big distinction one needs to make between him telling you that a specific coin came up heads, and that either one of them did.
This was actually part of the original question:
Your friend tosses two coins, then asks you to guess how the coins landed. You reply that you cannot know. Then your friend reveals that one of the coins he threw landed heads. Now how did the other coin land?
(Correct) answer: There is a 67% chance that the other coin landed tails. Why? Because after he told you one was heads, the remaining possibilities of the coin lands were heads-heads, heads-tails and tails-heads, therefore in two out of three cases the other coin is tails. If he told you that the first coin thrown was heads, then the chances would be 50-50, since there would be only two possibilities, heads-heads and heads-tails.
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On July 16 2011 07:37 Bibdy wrote:Show nested quote +On July 16 2011 07:14 hypercube wrote:On July 16 2011 02:38 Chill wrote:
Random:
HH x
HT o
TH o
TT x
50%
After knowing at least one was heads:
HH x
HT o
TH o
TT
66.6%
O_o
I think you're just thinking about it wrong. His knowledge isn't affecting the outcome - the outcome was random but he gave you specific information about the result that eliminates one of the possibilities.
If he flipped a coin, told you it was heads, and then flipped the other coin, there would be a 50% chance one of them was tails. But that's not the same thing. In scenario 1, TT was an option was that later eliminated with information. In scenario 2, TT is never a possibility so it doesn't factor in. How about this: Random: HH -> H HT -> T TH -> T TT -> H 50% After knowing at least one was heads: 1 x HH -> H 0.5 x HT -> 0.5T 0.5 x TH -> 0.5T TT50% I don't see a clear reason to prefer your distribution to mine. For example if he just looked at one of the coins in secret and told what he saw then the second distribution is the correct one (because for HT and TH he'd say tails 50% of the time). edit: to clarify, everyone who got 2/3 was assuming that every time one of the coins comes up heads your friend will say so. If that's true you are correct to just count up the cases. However, this assumes that he has an a priori preference for heads, i.e for HT he'll always say one of them is heads and never that one of them is tails. I just don't feel like that assumption is justified, given the description of the problem. I'm having a hard time figuring out why you're giving double the probability of HH occurring there. That's an assumption nobody made at all. The only assumption made is that your friend flipped two coins and you know that ONE of them came up heads.
It's called discounting in poker. There's an equal chance of the throw coming up HH or HT. But HH leaves your friend with only one option, he can only say that the one of the coins is heads. HT leaves him with two options: he can either say that one of the coins is heads or (presumably) that one of them is tails.
The chances of the throw coming up HT and him saying one of the coins is heads is 0.25x0.5=0.125 25% for HT and 50% for choosing to point out heads instead of tails.
edit: for HH we have 0.25x1=0.25. 25% for HH and 100% for pointing out heads. 0.25 is twice as much as 0.125 hence the "double probability"
The most important piece of information to realize in the entire puzzle is that your friend doesn't specify WHICH SPECIFIC coin came up Heads. Only that one of them did. Out of the four potential possibilities, the clue he gives you only rules out ONE of them, not two of them.
It rules out one but it might also discount TH and HT by 50%, depending on our model of his behaviour.
Hypothetically, if he told you that the FIRST coin came up heads, then your outcome list would look like this (remember that the first letter represents the first coin, and the second letter represents the second coin):
HH
HT
TH
TT
And you'd have a 50:50 chance, of picking the right answer. But, that's NOT what he told you. There's a big distinction one needs to make between him telling you that a specific coin came up heads, and that either one of them did.
Yeah, that's not what I mean.
This was actually part of the original question:
"Then your friend reveals that one of the coins he threw landed heads."
That's literally all we know about our friend. He tells the truth and he has the option to tell that one of the coins is heads. Is it a stretch to assume that he could say one of them is tails if HT or TH came up? I don't think it is.
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Coin 1 Coin 2
H1(.5) H(.5) = .25
H1(.5) T(.5) = .25
H2(.5) H(.5) = .25
H2(.5) T(.5) = .25
this is about as straight forward as it gets. H1 and H2 signify the two sides of the double-headed coin. as you can see, the probability of getting HH is 2/4, or 50%
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Okay, now you've both completely lost me. I don't see what's the point in modelling the guy's behaviour when its obviously a question that's meant to challenge your ability to consider how the information presented to you only rules out one possibility, nor can I figure out where the hell a double-headed coin comes from.
I'm just going to back away, slowly.
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On July 16 2011 08:29 Bibdy wrote:
Okay, now you've both completely lost me. I don't see what's the point in modelling the guy's behaviour when its obviously a question that's meant to challenge your ability to consider how the information presented to you only rules out one possibility, nor can I figure out where the hell a double-headed coin comes from.
I'm just going to back away, slowly.
If that's what you think there's no way I can convince you.
Won't bite on the double-headed coin.
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On July 16 2011 08:08 hypercube wrote:Show nested quote +On July 16 2011 07:37 Bibdy wrote:On July 16 2011 07:14 hypercube wrote:On July 16 2011 02:38 Chill wrote:
Random:
HH x
HT o
TH o
TT x
50%
After knowing at least one was heads:
HH x
HT o
TH o
TT
66.6%
O_o
I think you're just thinking about it wrong. His knowledge isn't affecting the outcome - the outcome was random but he gave you specific information about the result that eliminates one of the possibilities.
If he flipped a coin, told you it was heads, and then flipped the other coin, there would be a 50% chance one of them was tails. But that's not the same thing. In scenario 1, TT was an option was that later eliminated with information. In scenario 2, TT is never a possibility so it doesn't factor in. How about this: Random: HH -> H HT -> T TH -> T TT -> H 50% After knowing at least one was heads: 1 x HH -> H 0.5 x HT -> 0.5T 0.5 x TH -> 0.5T TT50% I don't see a clear reason to prefer your distribution to mine. For example if he just looked at one of the coins in secret and told what he saw then the second distribution is the correct one (because for HT and TH he'd say tails 50% of the time). edit: to clarify, everyone who got 2/3 was assuming that every time one of the coins comes up heads your friend will say so. If that's true you are correct to just count up the cases. However, this assumes that he has an a priori preference for heads, i.e for HT he'll always say one of them is heads and never that one of them is tails. I just don't feel like that assumption is justified, given the description of the problem. I'm having a hard time figuring out why you're giving double the probability of HH occurring there. That's an assumption nobody made at all. The only assumption made is that your friend flipped two coins and you know that ONE of them came up heads. It's called discounting in poker. There's an equal chance of the throw coming up HH or HT. But HH leaves your friend with only one option, he can only say that the one of the coins is heads. HT leaves him with two options: he can either say that one of the coins is heads or (presumably) that one of them is tails. The chances of the throw coming up HT and him saying one of the coins is heads is 0.25x0.5=0.125 25% for HT and 50% for choosing to point out heads instead of tails. edit: for HH we have 0.25x1=0.25. 25% for HH and 100% for pointing out heads. 0.25 is twice as much as 0.125 hence the "double probability" Show nested quote +The most important piece of information to realize in the entire puzzle is that your friend doesn't specify WHICH SPECIFIC coin came up Heads. Only that one of them did. Out of the four potential possibilities, the clue he gives you only rules out ONE of them, not two of them. It rules out one but it might also discount TH and HT by 50%, depending on our model of his behaviour. Show nested quote +Hypothetically, if he told you that the FIRST coin came up heads, then your outcome list would look like this (remember that the first letter represents the first coin, and the second letter represents the second coin):
HH
HT
TH
TT
And you'd have a 50:50 chance, of picking the right answer. But, that's NOT what he told you. There's a big distinction one needs to make between him telling you that a specific coin came up heads, and that either one of them did. Yeah, that's not what I mean. Show nested quote +This was actually part of the original question:
"Then your friend reveals that one of the coins he threw landed heads." That's literally all we know about our friend. He tells the truth and he has the option to tell that one of the coins is heads. Is it a stretch to assume that he could say one of them is tails if HT or TH came up? I don't think it is.
I'm not sure where you get the option for the friend from. The problem explains a very specific scenario that has occurred and has nothing to do with the friends behavior.
Here:
I have in the real world flipped two coins while posting at Teamliquid. One of the coins has landed heads. Now, what is the chance the other coin landed tails? This is reality, these coins are sitting in front of me right now. There is no other information. How do you determine the chance in this single scenario?
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He can't be this ignorant, he has to be playing devil's advocate at this point. The question he is discussing is losing all relevance to whats ACTUALLY being discussed, and its becoming another analogy that relates to a different scenario in order to make this logic make sense.
Math trolls. =/
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On July 16 2011 08:08 hypercube wrote:Yeah, that's not what I mean. Show nested quote +This was actually part of the original question:
"Then your friend reveals that one of the coins he threw landed heads." That's literally all we know about our friend. He tells the truth and he has the option to tell that one of the coins is heads. Is it a stretch to assume that he could say one of them is tails if HT or TH came up? I don't think it is.
Yes, it's a very big stretch. This is a math problem, not a real-life situation. The (implied) assumption in the original problem is that if at least one coin is heads, your friend will 100% of the time say that one coin is heads. Since we are considering a case where he does say one coin is heads....
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This is actually a problem with language and the framing of the problem. Your interpretation of the statement, "One of the coins is a head" determines the probability of the second coin being heads or tails.
If the statement being evaluated is, "Is at least one of these coins a head?" and the coin-flipper will always either answer, "One of the coins is a head," or, "It is not the case that one of the coins is a head," then the answer is that there is a 2/3 chance that one of the coins is a tail (I'd say it's misleading to say "the other" coin is a tail because a "first" or "original" coin was never specified).
If the statement the coin-flipper is answering is instead, "What side is showing on the first coin I examine?" then the answer is 50%.
For more information, see this article: http://en.wikipedia.org/wiki/Boy_or_Girl_paradox
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United States10328 Posts
Hi dudes, I'd like to quote myself from earlier in the thread (in case you didn't read it?)
On July 16 2011 03:12 ]343[ wrote:
OP: unfortunately your "counterexample" is different from the original problem. Instead, it would go like this:
You have two fair coins, each with a heads and a tails side. You aren't allowed to flip 2 tails: that is, whenever you flip two tails, flip both again. The probability you will get 2 heads now is indeed 1/3.
This is what it means when we're saying "at least one of the coins turns up heads": the possibility TT is thrown out. Nothing else has changed.
EDIT: Compare to what the guy above me said: we don't pick ONE of the coins and say "oh this is heads." All we are doing is throwing out the case when both are tails.
We assume the coins are distinguishable, so H1T2 is different from H2T1. But H1H2 = H2H1... since both coins are heads.
In fact, to make it easier, we can impose an ordering on the coins being flipped. The first coin will come up H or T, the second coin will come up H or T, but not both (so no T1T2). Notice that the first coin always comes before the second coin, so there's no such thing as H2H1. So the possibilities are H1H2, H1T2, T1H2, and the probability both are heads is 1/3.
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On July 16 2011 11:05 crate wrote:Show nested quote +On July 16 2011 08:08 hypercube wrote:Yeah, that's not what I mean. This was actually part of the original question:
"Then your friend reveals that one of the coins he threw landed heads." That's literally all we know about our friend. He tells the truth and he has the option to tell that one of the coins is heads. Is it a stretch to assume that he could say one of them is tails if HT or TH came up? I don't think it is. Yes, it's a very big stretch. This is a math problem, not a real-life situation. The (implied) assumption in the original problem is that if at least one coin is heads, your friend will 100% of the time say that one coin is heads. Since we are considering a case where he does say one coin is heads....
Except that assumption isn't in the problem, which makes it ambiguous. It's not the only possible assumption and different assumptions will lead to different results. Read the Analysis of ambiguity section in Roketha's link, it discusses exactly this question.
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On July 16 2011 11:45 ]343[ wrote:Hi dudes, I'd like to quote myself from earlier in the thread (in case you didn't read it?) Show nested quote +On July 16 2011 03:12 ]343[ wrote:
OP: unfortunately your "counterexample" is different from the original problem. Instead, it would go like this:
You have two fair coins, each with a heads and a tails side. You aren't allowed to flip 2 tails: that is, whenever you flip two tails, flip both again. The probability you will get 2 heads now is indeed 1/3. This is what it means when we're saying "at least one of the coins turns up heads": the possibility TT is thrown out. Nothing else has changed.
That's a huge assumption. What you're doing is that you are rewriting the problem, introducing new information and then go on to prove that the answer is 2/3. It is in your problem. But it's a different problem, so it doesn't help us.
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United States10328 Posts
On July 16 2011 11:57 hypercube wrote:Show nested quote +On July 16 2011 11:45 ]343[ wrote:Hi dudes, I'd like to quote myself from earlier in the thread (in case you didn't read it?) On July 16 2011 03:12 ]343[ wrote:
OP: unfortunately your "counterexample" is different from the original problem. Instead, it would go like this:
You have two fair coins, each with a heads and a tails side. You aren't allowed to flip 2 tails: that is, whenever you flip two tails, flip both again. The probability you will get 2 heads now is indeed 1/3. This is what it means when we're saying "at least one of the coins turns up heads": the possibility TT is thrown out. Nothing else has changed. That's a huge assumption. What you're doing is that you are rewriting the problem, introducing new information and then go on to prove that the answer is 2/3. It is in your problem. But it's a different problem, so it doesn't help us.
I'm pretty sure "at least one of the coins turns up heads" is mathematically precise. It means that both are not tails.
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Calgary25977 Posts
lol this thread is a fucking train wreck. Jesus christ.
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On July 16 2011 12:28 ]343[ wrote:Show nested quote +On July 16 2011 11:57 hypercube wrote:On July 16 2011 11:45 ]343[ wrote:Hi dudes, I'd like to quote myself from earlier in the thread (in case you didn't read it?) On July 16 2011 03:12 ]343[ wrote:
OP: unfortunately your "counterexample" is different from the original problem. Instead, it would go like this:
You have two fair coins, each with a heads and a tails side. You aren't allowed to flip 2 tails: that is, whenever you flip two tails, flip both again. The probability you will get 2 heads now is indeed 1/3. This is what it means when we're saying "at least one of the coins turns up heads": the possibility TT is thrown out. Nothing else has changed. That's a huge assumption. What you're doing is that you are rewriting the problem, introducing new information and then go on to prove that the answer is 2/3. It is in your problem. But it's a different problem, so it doesn't help us. I'm pretty sure "at least one of the coins turns up heads" is mathematically precise. It means that both are not tails.
It's also not from the OP.
"Then your friend reveals that one of the coins he threw landed heads."
This is the problem statement. The interpretation that he could have said "one of the coins landed tails" if he got TH or HT is reasonable. Others agree with me, read the boy or girl paradox article on wikipedia if you don't believe me.
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I feel bad for Chill....he is trying so hard to explain it to him lol!!!!!
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On July 16 2011 14:25 hypercube wrote:Show nested quote +On July 16 2011 12:28 ]343[ wrote:On July 16 2011 11:57 hypercube wrote:On July 16 2011 11:45 ]343[ wrote:Hi dudes, I'd like to quote myself from earlier in the thread (in case you didn't read it?) On July 16 2011 03:12 ]343[ wrote:
OP: unfortunately your "counterexample" is different from the original problem. Instead, it would go like this:
You have two fair coins, each with a heads and a tails side. You aren't allowed to flip 2 tails: that is, whenever you flip two tails, flip both again. The probability you will get 2 heads now is indeed 1/3. This is what it means when we're saying "at least one of the coins turns up heads": the possibility TT is thrown out. Nothing else has changed. That's a huge assumption. What you're doing is that you are rewriting the problem, introducing new information and then go on to prove that the answer is 2/3. It is in your problem. But it's a different problem, so it doesn't help us. I'm pretty sure "at least one of the coins turns up heads" is mathematically precise. It means that both are not tails. It's also not from the OP. "Then your friend reveals that one of the coins he threw landed heads." This is the problem statement. The interpretation that he could have said "one of the coins landed tails" if he got TH or HT is reasonable. Others agree with me, read the boy or girl paradox article on wikipedia if you don't believe me.
Thanks for that, I knew something wasn't sitting right with me.
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United States10774 Posts
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If my friend tells you that you don't know the laws of probability, what's the chance that you'll still be beating my GSTL fantasy team by 25 points?
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Ok I didn't read the whole thread but I take it that your main objection is the "unintuitive" nature of counting
HT
TH
as separate cases?
Well think of it this way: if you flip two coins the chances of getting exactly one head is twice that of getting two heads (if you don't buy this I suggest you try flipping some coins lol)
Ok now we only have two cases:
1: One head one tail (probability 2/3)
2: Two heads (probability 1/3)
(why? because the probability of case 1 must be twice that of case 2, and the sum must equal 1. Or 2x+x = 1 and solve for x)
Now you see that case one has probability 2/3 
Or maybe another way, suppose I color one coin blue and one coin red (same as imposing an "ordering"). Your friend tells you that at least one coin has heads. Well then I think you would agree that we should consider two cases: blue head red tails, blue tails red heads. Now if the friend could only say whether the blue coin was heads, well then he could only speak half the time (just like monty hall problem).
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EPT
