interesting coin flip problem

[PaultheS]

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All coin flips are completely independent. Regardless of how many you intend to flip and and how many you have flipped already and there results, each coin flip will have a 1/2 chance of being heads or tails. This deals with the law of large numbers. We can't apply the law of large numbers to a scenario in which there are only 2 trials. The greater the amount of trials preformed, the lower and lower the standard deviation becomes. When tossing coins, no matter what, chances are 50-50.
Knowing that you're tossing two coins does not change that.

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Yes, all coin flips are completely independent, but flipping two coins in a row is not the situation that is being dealt with here.

The question is: I flip two coins. Given that at least one landed heads, what is the probability that one coin was tails?

This is a very different question and has nothing to do with the independence of repeated coin flipping trials.

[AaronBrown]

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When you're given information that reveals that "one coin was heads", you aren't supposed to assume that this information was arrived at via random sampling. It's just a given piece of information.

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The trouble with this view is the interpretation of the information is affected by how it is given. To interpret a signal you need to know not just the signal, but what the signal might have been in different circumstances. Remember Sherlock Holmes and the dog that didn't bark.

When a person tells you "one coin was heads," we need to know what she would have said if both were heads and if both were tails. So the information is subject to interpretation.

[PaultheS]

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When you're given information that reveals that "one coin was heads", you aren't supposed to assume that this information was arrived at via random sampling. It's just a given piece of information.

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The trouble with this view is the interpretation of the information is affected by how it is given. To interpret a signal you need to know not just the signal, but what the signal might have been in different circumstances. Remember Sherlock Holmes and the dog that didn't bark.

When a person tells you "one coin was heads," we need to know what she would have said if both were heads and if both were tails. So the information is subject to interpretation.

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Good post. This is probably the best answer to the original question.

[PrayingMantis]

In your post you say this:

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Your teacher is right, and exactly for the reasons he stated.

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And then, after claiming it is like the Monte-Hall problem, you say this:

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The answer -- Assuming that Monty would have shown a door and offered the contestant the swap in all circumstances, is that the contestant should swap. There's a 1/3 chance the Big Deal is behind door number 1, and a 2/3 chance it's behind door number 3.

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However, the thing with the original and unspecified Monte-Hall problem is that you don't know if the assumption above about Monte and his "duties" is correct at all. Without more specified information (like you find in more detailed verstions of the problem), you don't know whether Monte indeed shows you a door and lets you swap in all circumstances.

That's the reason why you can't simply say that his teacher is right. His teacher might be right or wrong, according to different interpretations of the information he gives. And without any particular assumptions, the answer looks more like 1/2 than 2/3, as you need fewer assumptions in order to get to the 1/2 answer.

[kelvin474]

Three things are equally likely to be the state of the two coins, HH, TH, HT. We agree on that.

100% of the time any of these 3 events occurs, it is correct that "one of the coins landed heads".

So, 1/3 of the time, he has flipped HH. 1/3 of the time he flipped HT, and 1/3 of the time he flipped TH.

That means that 2/3 of the time "at least one head occurred", there is also an occurrence of tails.

The answer is 2/3.

If you don't believe this, send me a PM with where we can bet on this. Preferably for $1000/trial.

[BoogerFace]

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This problem is kind of like the Monte Hall -- Let's Make a Deal problem.

In that show, contestents were given the opportunity to go for the "Big Deal" which was behind one of three Doors.

A contestant chooses Door Number 1. Monte (who knows where the loot really is) shows the contestant that the Big Deal was not behind Door Number 2, and then offers the contestent the chance to change his choice of doors (i.e., swap door 1 for door 3). What should the contestant do.

The answer -- Assuming that Monty would have shown a door and offered the contestant the swap in all circumstances, is that the contestant should swap. There's a 1/3 chance the Big Deal is behind door number 1, and a 2/3 chance it's behind door number 3.

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And that's why the Monte Hall problem is stupid. The people who switch are only right 1/2 of the time.

[PrayingMantis]

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The answer is 2/3.

If you don't believe this, send me a PM with where we can bet on this. Preferably for $1000/trial.

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I'm quite amazed by the inability of people who are posting on a gambling forum to see that there could be few interepretations to the information given in the original post. Betting on "2/3" like you seem be willing to do is really a poor decision IMO for the simple reason that the conditions of the bet are not clear, and you are at the mercy of the person who decides how to manipulate the information he gives you. Why is that so difficult to understand?

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[KenProspero]

I think you miss the point, both the coin flip problem and the monte hall problem are probablility problems, not psychology problems.

The problem is designed to determine if you understand how probability works, and the assumptions which lead to the 2/3 answer in the coin flip, and "who knows what monte is thinking" in Monte Hall are not part of the probablistic analysis.

Further, in the OP, it was stated that the answer to the coin flip problem was 50-50 because they are independent events. At the time he posted, OP apparantly didn't understand why 2/3 was a valid answer.

Now, after reading this thread, if OP wants to go back and argue semantics with his/her prof, go right ahead. However, my academic experience is that with many profs, this would be a -ev exercise. Better in this case to understand the principle that the problem is trying to get at than to argue the nits.

Now, would I make a bet on this if someone offered me. Nope, if I did, I'm sure I'd end up with an earful of cider (Reference to Damon Runyon for anyone who didn't pick it up).

Some very excellent posts. Consider the 2 sibling boy/girl problem. Say there are 24 mothers who have two children. 6 of them have two boys, 12 have one boy and one girl, and 6 have two girls. They all come up to you and tell you the gender of one of their kids. Obivously, the 6 mothers who have two boys tell you "One of my children is a boy." The six mothers who have two girls say "One of my children is a girl." If you want it to be true that P(G|there is one boy) = 2/3, then you must assume that the twelve mothers who have both a boy and a girl tell you about their boy. This means that 18 mothers tell you "One of my children is a boy." In 12 of those cases, the other child is a girl. However, we also want it to be true that P(B|G) = 2/3. For that to be true, all 12 mothers must chose to tell us about their girl. Clearly, all 12 mothers can't chose to tell us about their girls and their boy. Thoughts?

[PrayingMantis]

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I think you miss the point, both the coin flip problem and the monte hall problem are probablility problems, not psychology problems.

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I never claimed that these are "psychology problems". But if you think these are "probability problems", in the purest sense, it seems to me that you don't fully understand both of these problems.

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The problem is designed to determine if you understand how probability works, and the assumptions which lead to the 2/3 answer in the coin flip, and "who knows what monte is thinking" in Monte Hall are not part of the probablistic analysis.

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No. You assume that "the problem is designed to determine if you understand how probability works", but you have little ground to base this on. With regard to the Monte-Hall problem, originally it was clearly NOT designed to determine if you understand how probability works, but rather to see if there's any advantage in switching doors. And I repeat - in the original unspecified Monte-Hall problem, there's no one simple solution, because you HAVE to know what Monte is thinking and doing in order to get to any reasonable solution, as opposed to what you suggest above. The probabilites on such a problem cannot be worked out in a vacuum.

There's a lot of work done with regard to the Monte-Hall problen, and I suggest you read some of it. The version of Monte-Hall problem you are thinking about (the one where the solution is 2/3) is not the interesting one, as that one is very easy to understand and solve.

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Further, in the OP, it was stated that the answer to the coin flip problem was 50-50 because they are independent events. At the time he posted, OP apparantly didn't understand why 2/3 was a valid answer.

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Reading the OP, It is clear to me that the poster understood why 2/3 is a valid answer. It is still a valid answer, BTW, but not the only valid answer.

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Now, after reading this thread, if OP wants to go back and argue semantics with his/her prof, go right ahead. However, my academic experience is that with many profs, this would be a -ev exercise.

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This is not "semantics", in the sense that understanding the nature of the information you have is crucial to solving the problem. In certain types of probability problems, the exact wording is very important, as opposed to most other kinds of problems. There are many examples for this.

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Now, would I make a bet on this if someone offered me. Nope, if I did, I'm sure I'd end up with an earful of cider (Reference to Damon Runyon for anyone who didn't pick it up).


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Examine this (note: this is not the original problem, but a version of it):

I have just flipped 2 unbiased coins, wrote the results and put in an envelope. I gave the envelope to a very respected poster here, that we both trust.

Now I'm telling you that one of the two results is heads. this result has a little star right next to it, in the envelope. I'm offering you 5:1 on a $1000 bet that the other result is tails. That is, if it's tails I pay you $5000, if it's heads you pay me $1000. That's the information you have, that's the bet I'm offering you. No further questions.

Would you (or anyone who thinks the answer to the first problem is definitely "2/3") take it, yes or no? And if not, why?