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Chris
guest
Jan-18-07, 04:14 PM (EST)

"Bear cubs problem"

 Hi,Great site!I know that spotting errors in probability puzzles is like looking for for water in the desert - most of the 'errors' are mirages which disappear upon closer inspection. However, I am going to stick my neck out and say that I think I have spotted an error!It is in the "Bear cubs problem" at https://www.cut-the-knot.org/bears.shtmlI think the error is in step 2. Now assume I told you that one of the bears is male. What is the probability that both are males? Of the three possible outcomes (mf, fm, mm) only the last where both bears are male is favorable. The answer is 1/3.All of step 2. is correct up to the very last line but it goes of the rails just a little bit before. Let me give you my version of step 2. Now assume I told you that (a particular) one of the bears is male. What is the probability that both are males? There are three possible outcomes (Wm Df, Wf Dm, Wm Dm) where W=White, D=Dark only the last where both bears are male is favorable. The last outcome can been true in two ways; Wm is the one male; Dm is the one male The other two outcomes can only be true in one way each. Thus, there can be four ways in which one bear is a male, two of which result in two males. The answer is 1/2.This then matches the probability for step 3.The puzzle as it'stands is very interesting because each step seems perfectly reasonable in its own right, yet, as you suggest, one intuitively thinks that specifying that the white bear is male shouldn't make any difference.However, we assume our error in thinking lies in step 3 and yet that step is perhaps the the clearest of all. Where as in fact if we re assess step 2. then the paradox is resolved.I hope this is right!

alexb
Charter Member
1951 posts
Jan-18-07, 04:33 PM (EST)

1. "RE: Bear cubs problem"
In response to message #0

 >Great site! Thank you.>It is in the "Bear cubs problem" at >https://www.cut-the-knot.org/bears.shtml >>I think the error is in step 2. >> Now assume I told you that one of the bears is male. > What is the probability that both are males? > Of the three possible outcomes (mf, fm, mm) > only the last where both bears are male is favorable. > The answer is 1/3. >>All of step 2. is correct up to the very last line but it >goes of the rails just a little bit before. Let me give you >my version of step 2. >> Now assume I told you that (a particular) one of the >bears is male. But the assumptions in the two cases are different. "One of the bears is male" and "A particular bear is mail" lead to different probabilities. With your assumption, the other bear is as likely to be male as it is to be female. With my assumption, there is in fact no "other bear" and one has to consider three possibilities.

Chris
guest
Jan-19-07, 11:15 PM (EST)

2. "RE: Bear cubs problem"
In response to message #1

 Slippery or what! You have caused me to think quite hard about some really basic meanings, only to end up spotting the obvious.It'seems to come down to what the word "one" is used to mean in that context. I agree with you that the assumption means (I hope) "we can count at least one male bear" and that the puzzle is correct as it'stands.However, we often use the word "one" to mean "a particular" e.g. "I've got a splinter in one of my fingers." Nobody would mean "I've got at least one splinter but it is unknown which finger it is in."It really is very interesting (to me anyway) the distinction between the two, which is in fact the essence of the conundrum in the first place.It also seems that it is tied up with how we know that one of the bears is a male.If we know one of the bears is male because the honey is gone from the top shelf and only male bears can reach that high, then there is a 1/3 possibility of 2 males.If Fred once saw one of the bears and never saw the other one and it was a male, then there is a 1/2 possibility of 2 males.The real difference, it seems to me, is how many bears are involved in providing the evidence for the statement "one of the bears is a male." If it is 2 bears, en masse as it were, as with the honey, then we are in the realm of the puzzle. If it is one bear, as with Fred, and in my miss interpretation of the puzzle, then it is 1/2 and we are already really at step 3.This is heart of the confusion that one instinctively feels in the first place - we interpret "one" as "that one".Sorry to ramble on so much and thank you for taking the time to rely.

alexb
Charter Member
1951 posts
Jan-20-07, 04:12 PM (EST)

3. "RE: Bear cubs problem"
In response to message #2

 You gave an excellent explanation of the ambiguity exploited by the problem. This kind of (language) ambiguities pops up all the time in common circumstances. I gave quite a few examples at https://www.cut-the-knot.org/language/index.shtmland the attached articles, especially in the correspondence with R. Hersh. >This is heart of the confusion that one >instinctively feels in the first place - >we interpret "one" as "that one".Perhaps you are right about that, but believe me, when I first came across that problem, my interpretation was different. It was exactly like that intended by a detective who claims that one of the suspects is in fact a culprit but, so far, it is unknown which one it is.

junglemummy
Member since Nov-7-05
Jan-24-07, 07:34 AM (EST)

4. "RE: Bear cubs problem"
In response to message #3

 The whole problem contains another, possibly invalid, assumption, ie that there is an equal chance that a random bear will be a particular gender. In this morning's paper, I read that there are 96 900 more women than men in Australia. So if you pick a random Australian, it is more likely that they will be female. Is there any gender weighting like this in bear populations? Perhaps the question should be prefaced with a statement like "in a zoo, there are equal numbers of male and female bears".

alexb
Charter Member
1951 posts
Jan-28-07, 07:55 AM (EST)

5. "RE: Bear cubs problem"
In response to message #4

 The implicit assumption is impractical but valid as any other assumption might be that is not based on scientific or statistical evidence. Even if the problem was about Australian bears or Arctic kangaroo it would make sense. The assumption of equal probabilities is implicit because it is the simplest that makes the problem interesting. All others would probably require an extra explanation which would divert attention from the essence of the problem.