EPTCoins tosses - Page 8
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Enhancer_
Canada320 Posts
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Cambium
United States16368 Posts
On July 16 2011 14:25 hypercube wrote: 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. I read your post about thirty times, and wavered between whether I should agree with you. My conclusion: no, this problem has nothing to do with ambiguous protocol. The problem states one specific instance with a defined protocol, and asked for the probability of the other coin for this particular instance. It seems to me that you have taken game theory, so in game theory terms, the state space is well defined. This is different from the hypothetical future where your friend would be throwing coins. He already threw the coins. Also, your claim that he could reveal a T is also false because the only ambiguity lies within the case of what would happen if both coins land T. The friend would never reveal T if the coins land HT or TH. | ||
frogrubdown
1266 Posts
On August 11 2011 06:29 Chill wrote: True. TT TH HT HH 2/4 True. TH HT HH 2/3 True. TT TH HT 2/3 True. XX XY YX 2/3 False. All options are still possible. If it lands TT, he will say "One of them is tails." If it lands HH, he will say "One of them is heads." Because he gets to choose, we can't eliminate any options. TT TH HT HH 2/4 True. This is actually the same as question 3 and 4 but you've worded it strangely. True. Because the rules are the same as question 3 and 4 (fair and random), knowing the rules doesn't change anything. All seems to be good here except for your interpretation of 6, which has stranger rules than I think you realize. Suppose you are told that one of the coins is heads. This will happen 50% of the time. 25% of the time the coins will land HH, every time of which you will be told that one coin landed heads. So 25% of all cases will be HH in which you are told one landed heads. 50% of the time (overall, not just in cases told heads) it will land HT or TH, and for each you will be told 50% of the time that one landed heads. So, 25% of all cases will be a HT or TH in which you are told that one of the coins landed heads. Therefore, exactly half of all cases in which you are told that one of the coins is heads will be cases in which the other coin is heads. There are twice as many rolls in which the two land differently, but these are reported as one landing heads half as often as when both land heads, evening the two situations out. So in scenario 6, being told one coin landed heads (or tails) doesn't actually give you any additional information. Since you aren't told the rules, it might still be rational for you to bet as though it did give you the usual additional information, but that seems tangential to the point of the thread. | ||
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frogrubdown
1266 Posts
On August 14 2011 08:18 frogrubdown wrote: All seems to be good here except for your interpretation of 6, which has stranger rules than I think you realize. Suppose you are told that one of the coins is heads. This will happen 50% of the time. 25% of the time the coins will land HH, every time of which you will be told that one coin landed heads. So 25% of all cases will be HH in which you are told one landed heads. 50% of the time (overall, not just in cases told heads) it will land HT or TH, and for each you will be told 50% of the time that one landed heads. So, 25% of all cases will be a HT or TH in which you are told that one of the coins landed heads. Therefore, exactly half of all cases in which you are told that one of the coins is heads will be cases in which the other coin is heads. There are twice as many rolls in which the two land differently, but these are reported as one landing heads half as often as when both land heads, evening the two situations out. So in scenario 6, being told one coin landed heads (or tails) doesn't actually give you any additional information. Since you aren't told the rules, it might still be rational for you to bet as though it did give you the usual additional information, but that seems tangential to the point of the thread. Not sure why I said all of the others are fine. The problem with 6 affects 7 in a fairly self evident way. Also, I think I can make my point clearer with some help from Reverend Bayes. Let A=You are told that there is at least one heads. The priors are as follows. P(A)=0.5 (we are given this in scenario 6) P(HH)=P(HT)=P(TH)=P(TT)=0.25 (that's how coins work) P(A|HH)=1 (we are given this) P(A|HT)=P(A|TH)=0.5 (same) P(A|TT)=0 (same) By Bayes' Theorem: P(HH|A)=P(A|HH)*P(HH)/P(A)=1*0.25/0.5=0.5 P(HT|A)=P(A|HT)*P(HT)/P(A)=0.5*0.25/0.5=0.25 P(TH|A)=P(A|TH)*P(TH)/P(A)=0.5*0.25/0.5=0.25 Therefore, P(HH|A)=P(HT|A) + P(TH|A). That is, the probability that the other coin is heads given that you are told one coin is heads is equal to the probability that the other coins is tails given that you are told one is heads. Please, no one comment on this point without reading what it is a response to and recognizing that the hypothetical is a slightly different one than the one in the OP. | ||
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hypercube
Hungary2735 Posts
On August 11 2011 14:24 Cambium wrote: I read your post about thirty times, and wavered between whether I should agree with you. My conclusion: no, this problem has nothing to do with ambiguous protocol. The problem states one specific instance with a defined protocol, and asked for the probability of the other coin for this particular instance. It seems to me that you have taken game theory, so in game theory terms, the state space is well defined. This is different from the hypothetical future where your friend would be throwing coins. He already threw the coins. Also, your claim that he could reveal a T is also false because the only ambiguity lies within the case of what would happen if both coins land T. The friend would never reveal T if the coins land HT or TH. I don't really want to go into too much detail, but I want to point out that your friend could hustle you by offering his $2 against your $3 in a bet to guess the second coin. Naively this bet has a positive expectation of (2/3)*2-(1/3)*3=1/3 dollars per bet. But in reality, even using fair coins, your friend can manipulate the frequency of heads to exactly 50%. He could do it in a variety of ways, including by alternating saying "one of the coins is heads" and "one of the coins is tails" or just by not offering the bet at all after looking at the coins. In this case your expectation would be negative, of course. edit: You could of course disallow all of these actions under the terms of the bet. My point is that you would have to be very specific to avoid getting hustled. In the end some people think the OP implies that the friend will allways say "one of the coins is heads" when it is. Maybe. I don't think there's much point in debating it, as long as it's clear where the disagreement lies. Which it is in your case. It's not a disagreement of logic or math, just interpretation. | ||
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hypercube
Hungary2735 Posts
On August 11 2011 06:29 Chill wrote: True. This is actually the same as question 3 and 4 but you've worded it strangely. True. Because the rules are the same as question 3 and 4 (fair and random), knowing the rules doesn't change anything. You are mistaken. If you accept the bolded part (in the first spoilered quote) as true the correct answer becomes 1/2. Most people who argue for 2/3 don't accept it and implicitly or explicitly assume that for one heads one tails you are always told "that one of the coins he threw landed heads". | ||
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SpoR
United States1542 Posts
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