A Statistics Question for Nick Matzke

_{Barry Arrington

December 16, 2013

Intelligent Design}

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If you came across a table on which was set 500 coins (no tossing involved) and all 500 coins displayed the “heads” side of the coin, would you reject “chance” as a hypothesis to explain this particular configuration of coins on a table?

Comments

I like #148 too, but it's not the original problem. from #145:
“Mr. Brown has exactly 2 children”, there are 3 equal sets: A: (a set of) 2 boys B: (a set of) 2 girls C: (a set of) 1 girl & 1 boy
You can use 3 sets but they won't be equal. The mixed pair will happen twice as often as either 2 boys or 2 girls. The coin toss example is more intuitive for this - flip a coin twice and probabilities are: 25% both heads 25% both tails 50% combination of heads and tails.Piltdown2_{December 19, 2013
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#148 I think is the right answerc hand_{December 19, 2013
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It is ambiguous as to which child is a boy. It could be child1, child2 but not both at the same time and in the same sense.
That would be true if ONLY one child were a boy, if at least one child is a boy, then both children can be boys.scordova_{December 19, 2013
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Mr. brown has two kids. The one in college is a boy, the other I don’t remember.
In this case there are only two possible outcomes. 1. college-child is a boy, non-college child is a boy 2. college-child is a boy, non-college child is a girl probability of the other child being a girl is 1/2 since one of the two possible outcomes involves the other child being a girl.scordova_{December 19, 2013
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It is ambiguous as to which child is a boy. It could be child1, child2 but not both at the same time and in the same sense.c hand_{December 19, 2013
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Do we need a sequential (?) distinction between the two children?
No. That's why it's better to talk coins even quarters and dimes. Now there is a subtlety, if I say at least one coin is heads, you don't know if I'm talking about the quarter, the dime, or both. That's why this problem is difficult. The problem used phrases we are not familiar with. I would never tell you in the course of normal conversation:
Mr. Brown has exactly two children. At least one of them is a boy
chances of the other child being a girl is 2/3 I'd say something like this
Mr. brown has two kids. The one in college is a boy, the other I don't remember.
Chance the other kid is a girl is 1/2. Why? Because in the course of normal conversation, Mr. Brown is making it clear which child he is talking about (i.e. Child1). It would be like Mr. Brown saying "the quarter is heads". The reason the original word problem is difficult to understand is that it is state din a very vague and confusing manner. That was the point of the exercise to see if one can actually sort out the truth from confusion. You almost never hear things stated in that way. Usually you'll here, "Mr. Brown has two kids, one kid in college, a boy, I don't know about the other." In that case the probability of the other child being a girl is 1/2 because now we have identified which child (the one in college) is definitely a boy. In the case of the original problem Mark posed, the is severe ambiguity as to which child is a boy. It could be child1, child2, or both. The problem is difficult because it is far removed from ordinary experience and ordinary language.scordova_{December 19, 2013
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Scordova #140: There are only 3 ways the above statement can be true: A. Child 1 is a boy, Child 2 is a boy B. Child 1 is a boy, Child 2 is a girl C. Child 1 is a girl, Child 2 is a boy Talking about the children’s ages and you meeting them is creating a different question than the one posed. The designation of Child1 has nothing to do with age, it’s just something to distinguish him/her form child2.
You say that ‘the designation of Child1 has nothing to do with age’. What then is the difference between B and C? If we discard age how does B and C constitute a different outcome? When we discard age why isn’t accurate to say: “Mr. Brown has exactly 2 children”, there are 3 equal sets: A: (a set of) 2 boys B: (a set of) 2 girls C: (a set of) 1 girl & 1 boy Do we need a sequential (?) distinction between the two children? Why not instead think of them as mathematical sets? A set of 100% boys, a set of 100% girls and a set of 50% boy & 50% girl.Box_{December 19, 2013
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c hand, I think I finally get this. You are doubling the probability of boy/boy. With first sentence, "Mr. Brown has exactly 2 children", there are 4 equal possibilities - bb,bg,gb,gg, and 3 out of the 4 have a girl. With new information that "at least one is a boy", you eliminate gg. That leaves the other 3 possibilities, which are still equal, 2 of which have a girl. So the probability of having at least one girl dropped from 3/4 to 2/3 (and the probability of having a boy rose from 3/4 to 3/3). If you identify a particular child as a boy (or a girl), it's a different scenario. In that case, you are left with 2 equal possibilities for the unknown child - either boy or girl, so the chance the unknown child is a girl is 1/2.Piltdown2_{December 19, 2013
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sal, There are only 4 ways the above statement can be true: A. Child 1 is a boy, Child 2 is a DIFFERENT boy B. Child 1 is a boy, Child 2 is a girl C. Child 1 is a girl, Child 2 is a boy D. Child 1 is a DIFFERENT boy, Child 2 is a boy Child one and child two CANNOT be the same boyc hand_{December 19, 2013
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Box: First, 500 coins on a table could be scattered every which way, so I shifted to in a row. Next, being "told" [presumably accurately] that we have all H, alternating H & T etc is a way of saying we are informed by a message. What I am trying to do is help you see why information is measured the way it is. KF PS: BTW, in the original post, explicitly tossing is ruled out.kairosfocus_{December 19, 2013
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MF: At this point, I seriously doubt that your problem is lack of clarity. KFkairosfocus_{December 19, 2013
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c hand, A more clear way of stating the original problem
Mr. Brown has exactly two children. At least one of them is a boy. What is the probability that the other is a girl? You did not meet any of the children, you don't know whether child 1 is a boy and you don't know whether child 2 is a boy. All you know is: Mr. Brown has exactly two children. At least one of them is a boy.
There are only 3 ways the above statement can be true: A. Child 1 is a boy, Child 2 is a boy B. Child 1 is a boy, Child 2 is a girl C. Child 1 is a girl, Child 2 is a boy Talking about the children's ages and you meeting them is creating a different question than the one posed. The designation of Child1 has nothing to do with age, it's just something to distinguish him/her form child2.scordova_{December 19, 2013
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When Mr. Brown introduces you to his son, you know that the missing sibling could be either a girl or a boy with 50/50 probabilities.
You are assuming that the way that you got the information about one child being a boy was by you meeting the boy yourself. That is not what is stated in the original problem. The original question:
Mr. Brown has exactly two children. At least one of them is a boy. What is the probability that the other is a girl?
Answer: 2/3 Your scenario is different, and I'll word it this way to hopefully get the point across:
Mr. Brown has exactly two children. You met one of them, and the one you met was a boy. What is the probability that the other child is a girl?
Answer : 1/2 (or using your term "50/50" ) In fact an even more minimal version:
Mr. Brown has exactly two children. You met one of them. What is the probability that the other child is a girl?
Answer : 1/2scordova_{December 19, 2013
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Box, I understand your point about the need for a sequence, temporal or spacial, in order for alternating or ASCII text specification to make any sense. All heads and nearly 50/50 states can be detected irrespective of any arrangement. The original OP used all heads as an uncontroversial (Yeah, right!) way of highlighting a state that should have been readily dismissed as due to chance alone. In that case the "order" of the coins was irrelevant. However, what is the purpose of discussing the heads/tails states 500 coins (as proxy for more interesting systems) if their only relationship is that they are in the same room? The more interesting systems typified by the 500 coins do have temporal or spacial relationships among their elements. Is it not those systems we are seeking to better understand? So, I think it is appropriate to examine the question with order of some sort within the coin set. If you disagree, so be it, but there is little to be learned in that case. StephenSteRusJon_{December 19, 2013
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SRJ #135: If I were to ask you the state of coin 47 in a set of 500 coins, (you know to be) all heads, you already have all the information you need to answer. If I were to ask for the state of coin 47 in a set of coins, (you know to be) alternating heads and tails, you could not tell me until you examined at least one, any one, of the coins. If I asked for the state of coin 47 in a completely unspecified (unknown states) set of 500 coins you would have to locate coin 47 and examine it to give me the correct answer since learning the state of any, or even all, of the other coins is of no help.
I agree with everything you have said, providing that there is a (tossing) sequence involved. Talking about "alternating heads and tails", like you do, only makes sense in the context of a tossing sequence. However, by assuming such a tossing sequence (or any sequence) are you not changing Barry's scenario - just like KF did? A sequence does not follow from the OP. From the OP we just know that there are 500 coins on the table and they all display the “heads” side of the coin. We do not know if they are lined up in a row or in which sequence the configuration appeared - if any. It doesn't make sense to say you have information about a sequence which may not be there.Box_{December 19, 2013
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Sal MF, There are two variables to apply to Mr. Browns children, known-unknown, and boy-girl. Since the known child is a boy we get possibilities of: known BOY - unknown Boy unknown Boy - known BOY known BOY - unknown GIRL unknown GIRL - known BOY It is obvious that the birth pattern can alternate with girl-boy, less obvious with boy - boy When Mr. Brown introduces you to his son, you know that the missing sibling could be either a girl or a boy with 50/50 probabilities.c hand_{December 19, 2013
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Box, I think you may be wading in too deep with the see vs told. Recognize that "seeing" and "having being told" simply mean "knowing". In other words, "having information." If I were to ask you the state of coin 47 in a set of 500 coins, (you know to be) all heads, you already have all the information you need to answer. If I were to ask for the state of coin 47 in a set of coins, (you know to be) alternating heads and tails, you could not tell me until you examined at least one, any one, of the coins. If I asked for the state of coin 47 in a completely unspecified (unknown states) set of 500 coins you would have to locate coin 47 and examine it to give me the correct answer since learning the state of any, or even all, of the other coins is of no help. By the way, the specification need not be as simple as "all heads" to "know" the state of coin 47 without examination. The specification could be "the ASCII representation of the first 72 characters of this text" (which may have a nearly 50/50 blend of apparently random bits) and you would "know" the state of coin 47 with out examination of any of the coins. Does that help? StephenSteRusJon_{December 19, 2013
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KF #128: Also, I did not speak of how we have exhaustive information on the coins, obviously we do not, we don’t even know the denomination, or the dates or the metals etc.
Of course, I would never go there. You are stating the obvious.
KF #128: What I spoke to is how, on knowing that one has 500 coins in a row — all H, one has sufficient information to know the relevant state in the context of H/T as defining state.
You are talking about a scenario in which we are told that there are 500 coins on the table and that they are all heads up. In this scenario we are told about the coins but we cannot see for ourselves. Now you say: we have enough information to know the exact sequence.
KF #110: The point is, if you are told, 500 coins all H, there is one possibility, this is very specific in the set of possibilities.
KF #128: Knowing that coins are alternating H and T gives less info, and knowing only that coins are 50:50 H and T gives even less.
Ok, clear, obvious, I understand what you are saying. But, why did you change Barry’s scenario into your own?
OP: If you came across a table on which was set 500 coins (no tossing involved) and all 500 coins displayed the “heads” side of the coin, would you reject “chance” as a hypothesis to explain this particular configuration of coins on a table?
In Barry’s scenario we are not told "500 coins all heads", instead we see the coins and their configuration directly ourselves. And if we are not told about the coins on the table there is no sentence “500 coins all H” which contains information about the configuration of the coins. Changing scenarios, without telling, causes a lot of confusion.Box_{December 19, 2013
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KF - I think it is best for both of us if I stop trying to understand and respond to your comments. It would only lead to tears.Mark Frank_{December 19, 2013
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MF: With all due respect I must say: there you go again. You have an extremely familiar situation -- dropping a die that comes to rest with a side uppermost. Something that I don't doubt that if you really wanted to you could scrounge around in your home and set up as an experimental exercise. However, I doubt that you need to do so as the matter is abundantly familiar to one and all from board games. Aspects of the situation -- dropping a die that falls, tumbles and rolls then settles to a reading -- have been brought out that distinguish the causal factors: 1 --> mechanical necessity, manifest in the falling 2 --> chance, manifest through the tumbling and settling (with an injection of the butterfly effect) 3 --> design, as would happen with a loaded die or by simply reaching over and setting the die to a required value. From these cases, we can identify to short definitions; and yet, you are off playing context-switching word games and tangent games again. I repeat, for record, that: a: mechanical necessity leads to reliable regularities of low contingency, b: chance denotes credibly undirected contingency, and c: design, denotes directed contingency. If you still don't understand the difference between b and c, think about driving a car down a country lane with an unpaved rugged surface. Intelligent steering is obviously different from keeping the hands off the wheel and letting the car bounce from one rut to the next. And the results will be quite different, too. At this point, what chance is and what design is, and what m4echanical necessity is, have been adequately explained for a reasonable person. I strongly suspect that at root, you don't believe that there is real free intent and decision making leading to design, as if you run true to the form of your announced worldview -- evolutionary materialism, you don't believe in a self-moved actuating cause that can freely make decisions and act on them. That is, you want in the end to implicitly reduce design to other factors. The problem with that being, once you write off freedom to choose and act, you write off freedom to reason, which undermines the whole project of intelligent thought and discussion. That is, reduction to absurdity. KFkairosfocus_{December 19, 2013
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Coldcoffee The key difference between a loaded die and a defective die is that the loaded die has been made to give non-uniform probability distribution intentionally while in the case of a defective die this has happened accidentally. For example, a dice might have an off-square face or an uneven weight distribution by accident or on purpose. However, it would be impossible to tell the difference by looking at a series of throws. It would be necessary to know something about the process of making the die. What I found confusing about KF’s example was that he offered a fair die as an example of non-directed contingency and a loaded die as an example of directed contingency (I took took “loaded” to mean “biased” – but I happy to be corrected and accept that “loaded” means intentionally biased). My impression was that KF was proposing: a) non-directed contingency corresponds to “chance” and directed contingency corresponds to “design”. b) it is possible to tell whether a series of throws is directed or non-directed by looking at the throws and without understanding the process by which the die was made. (However, I have the greatest difficulty understanding KF’s writing – so I may be wrong) However, a fair die is almost always intentionally designed and while a loaded die may, by definition, be designed a biased die resulting in the same series of throws may well be accidentally biased.Mark Frank_{December 19, 2013
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=> Mark Frank, If you take a fair die as a Discrete uniform distribution, various probabilities can be computed for fair value. For Eg the probability of getting a total greater than or equal to 10 in two dice roll is 0.167, the probabilty of getting greater than or equal to 10 in 3 dice roll is 5/8=0.625. So if there is deviation from the probability, we can say the dice is either defective or loaded. The loaded die will have a greater frequency of a PARTICULAR dice value than a merely defective die.coldcoffee_{December 19, 2013
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MF: Pardon, but with all due respect, the last comment at 115 is utterly revealing of rhetorical squid ink evasive gamesmanship:
MF: So a fair die is undirected contingency while a loaded die is directed contigency? This is really confusing. After all the fair die is designed to be fair and a loaded die may well be loaded as a result of a natural mishap.
FYI, if a die is defective it is defective, not LOADED. A LOADED die is one that has been manufactured or altered to be biased for a purpose. A fair die has been manufactured successfully to be fair, i.e. give a near flat distribution. And that a die is always a manufactured entity is irrelevant to the issues on the table. Here is Wiki, FYI -- or rather for the onlooker who actually evidently wants to deal seriously with the matter at stake instead of running away behind a cloud of rhetorical squid-ink:
A loaded, weighted or crooked die is one that has been tampered with so that it will land with a specific side facing upwards more or less often than a fair die would. There are several methods for creating loaded dice, including round faces, off-square faces and weights. "Tappers" have a mercury drop in a reservoir at the center, with a capillary tube leading to another reservoir at a side; the load is activated by tapping the die so that the mercury travels to the side. Another type of loaded die is hollow with a small weight and a semi-solid substance inside whose melting point is just lower than the temperature of the human body, allowing the cheater to change the loading of the die by applying body heat, causing the semi-solid to melt and the weight to drift down, making the chosen opposite face more likely to land up. A less common type of loaded die can be made by inserting a magnet into the die and embedding a coil of wire in the game table; running current through the coil increases the likelihood of a certain side landing on the bottom, depending on the direction of the current. Transparent acetate dice, used in all reputable casinos, are harder to tamper with than other dice. A die may be shaved on one side, making it slightly shorter in one dimension, thus affecting its outcome. One countermeasure employed by casinos against shaved dice is to measure the dice with a micrometer before playing.
Your attempt to artfully manufacture an error on my part by way of an evasive, context-switching red herring side track word game both fails the basic English test, and is a sign of willful manipulation. A fair die is indeed manufactured to be fair, that was never at stake, dice are artifacts. The EIGHT YEARS LONG context -- which you artfully tried to switch -- is that, first, a dropped heavy object falls reliably showing mechanical necessity. Then, if it is a die . . . notice no discussion of manufacture that is irrelevant at this point . . . it will tumble and settle to a reading that is a manifestation of high contingency. An extremely familiar example. One on the table for EIGHT years. Then, in that context we can introduce a distinction: that some dice are fair, and tumble in ways driven by the butterfly effect (aided by twelve edges and eight corners) that leads to an outcome that is unpredictable in the specific case, and in effect will take a flat random distribution value on the range 1 to 6. Which last implies that we are discussing common dice. Where also the butterfly effect is a popular way of saying what more specifically is sensitive dependence on initial and/or intervening conditions, which will magnify small initial or intervening variations to yield drastically dissimilar outcomes. The chaos phenomenon in short. Then in that context we consider another known source of high contingency, design. This can be by loading the die or by simply setting a die to read a given value. The squid ink cloud, rhetorical game now becomes utterly apparent:
a: confronted with a simple and familiar case, b: you have tried to find wedge-points to push it out of an obvious and familiar context [exploiting the fact that language is always used in a context], c: to find some Wittgenstein alternative context or game that would make the words used into something else d: that can now be cleverly deemed confusing or meaningless or ludicrous. And on the other hand, e: if there is an effort at giving enough technical or other details to specify the situation more exactly to avoid this, there is another resort: f: oh it is too long and complicated and confusing. I need not pay attention. _________________________ g: Heads I win, tails you lose, in short.
Games over. Back to the basic point, the first key concern in a causal process is high vs low contingency. Low contingency situations will reliably have essentially the same effect on similar initial circumstances and are characterised by mechanical necessity manifest in law-like regularities, e.g. classically F = m*a and deterministic dynamics expressed in more complex differential or difference equations. A dropped heavy object near earth's surface falling at 9.8 N/kg is a classic, highly familiar illustration. But there are situations that are highly contingent, where under similar initial circumstances, highly diverse outcomes are possible or observed. The tumbling of a dropped die and its settling to different readings is a familiar example, indeed a paradigm. (NB: This also takes in coins, the coin being a two-sided die.) That high contingency has two commonly known explanations: (i) undirected and (ii) directed contingency. The former is a more sophisticated description of what we call chance and is in the design inference filter the default explanation for high contingency. (There is a deliberate choice to err on the side of chance, in order to be conservative in e4stimateing that something is caused by design.) Directed contingency is also familiar, with a loaded die or a die simply set to a reading as simple and familiar illustrations. So, chance is sufficiently characterised for those who are serious as credibly undirected contingency as a causal factor or influence. By contrast, design is directed contingency, and mechanical necessity denotes low contingency, reliable patterns that are generally driven by law-like forces similar to falling under gravitational attraction. Yes, there can be elaborations and complexities, so that the design filter is applied on a per aspect basis, an aspect being a view of part of an overall situation isolated for analytical purposes. For classic instance if one does a pendulum exercise, one will see that there are patterns that fit in with law-like regularities, but also scatter around the trend line then at a certain point -- about 6 degrees of amplitude -- where the accuracy of the simple law usually given falls off, showing a built in bias of design. In this case, the simple pendulum law is limited to small amplitude swings. the scatter about the trend line is usually held to be a matter of chance factors disturbing the situation and its measurement, and often show a Gaussian curve. That is, a summation of many small positive or negative factors clustering around a mean in a familiar bell-pattern. We could elaborate this into a discussion on errors of observation, sources of scatter, personal equations and the like but that would be pointless. MF needs to understand that the analysis of causal factors seen above is based on well known physical phenomena and cases, some of them actually famous or historically important, e.g. systematic errors due to wear were important in the characterisation of the metre based on a major survey. KFkairosfocus_{December 18, 2013
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Box: Have you looked at the summary presentation on information theory which I linked? (This is a good slice of the basis for the modern information age that we are in, e.g. it is the root of the term, "bit.") Also, I did not speak of how we have exhaustive information on the coins, obviously we do not, we don't even know the denomination, or the dates or the metals etc. What I spoke to is how, on knowing that one has 500 coins in a row -- all H, one has sufficient information to know the relevant state in the context of H/T as defining state. Knowing that coins are alternating H and T gives less info, and knowing only that coins are 50:50 H and T gives even less. In the context of having a vocabulary or alphabet of possible signal states [H/T is an example, so is T/F, 1/), or A/B/C . . . or A/G/C/T or U . . . . ], it is possible to construct a metric of information carrying capacity, commonly called Shannon information. Where the Shannon info of symbols from such an alphabet, is the avg info per symbol based on a weighted average of info carried per symbol. Info being developed as a log of reciprocal probability metric: Is = - log ps, for various reasons. As was outlined here. KF PS: I note that design thought moves beyond the concept of Shannon info carrying capacity of symbols, to metrics developed in various ways that are tied to function based on specific configurations. For instance, consider a case of 500 coins spelling out in ASCII code, the first 72 or so characters of this message.kairosfocus_{December 18, 2013
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you have been placing boy-girl combinations in a birth order and counting each, but you have allowed for only one boy-boy birth orderc hand_{December 18, 2013
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the illusion is created by treating the boys or the head side of coin as interchangeable entities , while the girls or the tails side of the coin remain discrete. You don't need to stipulate the birth order of the boy, he is either the older or the younger, but he can't be both.c hand_{December 18, 2013
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C hand, I can appreciate that you are convinced the odds are 50/50 that the other coin is tails. That would have been true in the absence of other information such as "at least one coin is heads." But now we have more information and we must revise our odds accordingly in light of the information in order to make an accurate estimate of the probability of certain outcomes. There are 3 possible outcomes:
dH qH dH qT dT qH
Two of the outcomes have tails (the bolded ones). Thus two out of the 3 possible outcomes have tails thus the odds are two out of three or 2/3 of the time the other coin is tails. The problem is the English language and our intuition about the problem will distort our perception. The challenge isn't learning the problem, it's un-learning how we usually conceive of the problem. What I've provided are the odds as calculated in basic statistics and discrete math books. If you are sure I'm wrong, you can cross check the link Box provided (it appears Box may have changed his mind). The reason this example is emphasized in textbook math is students have great difficulty unlearning the way they think about such problems. Student instinctively think 50/50 odds as you have. The odds here are 2/3, just as Mark has said. Salscordova_{December 18, 2013
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sal, the two coins are not and can not be interchangeable. If you label one a dime the other a quarter you get: dH qH dH qT dT qH The first coin(which you see is heads) must either be the dime or the quarter, and you know that you are looking at 50/50 oddsc hand_{December 18, 2013
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Mr. Brown has exactly two coins. At least one of the coins is a heads. What is the probability that the other coin is tails?
How about this; no arithmetic involved: There is obviously no relation between the fact coin A is heads and the state of coin B. The knowledge that coin A is heads is therefore irrelevant. So the probability that the other coin is tails is 50%Box_{December 18, 2013
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Box, Yes indeed. I actually remembered something like Mark's question from statistics class a long time ago, but it was phrased in terms of coins. The class text pointed out even this simple example would frequently trip up the best students. So I didn't have to consult the net on that question because I had been already taught it but using coins instead. Looking back, I don't think I would have solved it without some serious pondering. Simple problems aren't always so simple are they? :-) Salscordova_{December 18, 2013
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