md65536

I've thought about it some more and figure that the behavior of light will be different, even in the case of a coasting universe. If a light source is receding due to velocity with no expansion of space, a photon approaching Earth will approach at c. If the source is receding due to expansion, the closing rate of a photon will be less than c (or negative if it's beyond an event horizon). True, but the redshift of a coasting galaxy should remain constant, so it should never be increasingly redshifted toward 0. The example I used earlier, of a galaxy receding at 2c but remaining at that recession rate, doesn't make any sense!, because it would have had to increase its rate of recession up to that rate, and then start coasting, and an explanation of how that could possibly be would then be required. I agree with you too, I was just curious about if it's even possible to reason about some of the alternatives others have posted (including some that contradict Krauss's predictions). I think the ant analogy would be fine as long as the rope is given the same properties as space, and the ant the properties of a photon. After some thought, I realize I don't quite know enough of the details related to this thread. Let's assume that space *is* expanding as has been observed, and that there is a cosmic horizon beyond which any events that occur will never be seen. Is the horizon at the same distance all around? (I think yes because space is homogeneous and expanding uniformly???) Is the distance to the horizon increasing over time? Or decreasing, or staying the same? For a given distant galaxy to be forever visible, the horizon would have to be receding fast enough that the galaxy never crosses it. If the horizon remains at a fixed distance over time, that would mean that space of that length is expanding at a fixed rate. This is the simplest case in my mind but I don't think it matches observations. If the horizon is approaching, then the rate of expansion per unit of length is increasing.

Do you agree that nothing would tell us this had happened, or do you think we'd have the ability to observe the difference?

I don't understand the meaning of expansion of space in such a case, because I don't see a difference between the galaxy coasting away from us, and the expansion of space coasting, in that case. If the rate of recession is not in any way a function of the distance between the two, then what makes it an expansion of space, rather than just a simple velocity between the two? It seems like an unnecessary complication to say that space is expanding, if that space has no bearing on the behavior of the recession of objects. Or in other words, if the universe were freely coasting, then what is the measurable distinction between an object receding due to expansion, and one receding due to velocity? ANYWAY, I'm digressing. Observations point to an accelerating expansion, and Krauss's conclusions assume that that is true and will continue, in which case there is a horizon and photons from events behind it will never reach us, even given an infinite passage of time. And, like previously conceded, if everything we know about the universe can turn out to be wrong, then any prediction we make can turn out to be wrong. Not understanding something is not enough justification to argue that it is wrong.

If c remains constant relative to their own metric, or local ruler, then the ruler is expanding at the same rate as space? What is the meaning of expansion if the ruler also expands? The size of the universe relative to the length of the ruler doesn't change. How do you measure expansion, if your measurement is the same before and after the expansion?

I don't think that's helpful, because your demonstration is not at all like what is observed happening in space. I know the science isn't settled, and who knows--perhaps what you describe could be like reality. But I don't think anyone in mainstream science thinks so. Imagine that space is expanding as you suggest, where our hypothetical galaxy is receding at a constant rate. Photons, or ants, can only complete their journey because the distance that they still have to go is expanding slower than it was before. For example, suppose that the galaxy is at a distance of x and receding at 2c. Suppose that after n years the photon/ant has made back to its original distance of x from Earth. Assuming (contrary to mainstream science) that the galaxy is still receding at 2c, and that the entire space between the galaxy and us is expanding uniformly, then the space between the photon/ant and us is expanding at a rate less than 2c. So the space from Earth to x used to be expanding at a rate of 2c, and now space across that distance is expanding at a lower rate. What interpretation of expansion of space makes that possible? Your example is like the expansion of a balloon, but not I think like predictions of expansion of space. My understanding of expansion of space is that all space is expanding uniformly. That means that if the galaxy is 10 LY away, and if that 10 LY is expanding at a rate of c, then after some amount of time when the galaxy is 20 LY away, each of the 2 10-LY sections between it and us will be expanding at a rate of c. This assumes there is no change in the rate of expansion per a given distance over that time. It implies exponential recession of the galaxy. So, even if the rate of expansion per LY is fixed, the galaxy accelerates its recession. Even if the rate of expansion per LY is decreasing over time, depending on by how much, it's possible for the galaxy to accelerate away from us. Do I have this wrong?

Yes Willa, I believe that answer is correct. Back to the original problem, I think it's time I try to provide an answer. When I made this puzzle I didn't realize the numbers were so ugly, they seemed small! The answer below is provided by Wolfram Alpha. When I first tried to solve it, I couldn't do sums of combinations because of a bug in Wolfram Alpha, but I submitted a report and they fixed it, so this puzzle might be the reason you can do sums of combinations in Wolfram Alpha! I figure it deserves to be answered. I answered this the way that I would if I had to work it out on paper, trying to minimize the number of calculations (the way I assume the average statistician would solve it in their head). This probably isn't the simplest solution. Also I might have made a mistake in the reasoning or the maths... EDIT: I did in fact make a mathematical error in the first attempt...

From a stats perspective the answer is extremely simple... It's an easier puzzle when you don't think about it! You can solve it many different ways, and the more complex solutions will work out the same as the simplest, due to the consistency of maths. By thinking about it, it becomes a brain teaser about intuitive (or unintuitive) aspects of stats, including the consequences of the independence of independent random variables, and also being convinced that they really are independent. A slight modification to the puzzle can change it, and make the variables not independent, which is probably a common source of confusion because when we try to think of real world examples, intuition of "how things should be" can sometimes add unnoticed biases.

If you do it this way then you also have to take into consideration the probability that a random selection of one will be B. That is 1.0, 0.5, 0.5 (and 0.0) respectively. Another way to achieve the same result is to take the first of the pair as the escaped cow. Like you allow, case WW doesn't happen. If you do it this way, neither does WB. I worded the puzzle to be sure not to imply that a brown cow was specifically selected for its color, to escape. In your example, what is the reasoning for choosing the first cow in the BW case, but the second in the WB case? Agreed; this is what I meant when I said that the knowledge of a brown cow should not make one think that there will be fewer than 50% brown cows. Try your examples with coin tosses. Suppose you toss a coin and it is heads. What will the next toss more likely be?

A paper by N. David Mermin --- "What's bad about this habit" http://www.ehu.es/aitor/irakas/mes/Reference/mermin.pdf --- mentioned previously on this site, is a good read on this topic. A lot of that comes down to interpretations, including what the measurements imply or what they mean. Another part of it has to do with assumptions, including about things like "what happens to stuff in between when it is measured?" Per the above article, it is a good habit to not confuse a model with reality. Another good habit would be to not have an inappropriate certainty about what exists, especially beyond what is measured. With a useful definition of existence, all of our knowledge of existence can be considered uncertain to varying degrees.

I meant the proportion of the 100 cows (the farmer still has 100 cows), but I wasn't very clear. But still, the answer you gave isn't correct for the proportion of cows in the barn either. It it essentially a random cow that escapes, which happens to be brown. It is not selected to be brown (see Escaped cow puzzle 2 for that version!). The color of each cow is independent, meaning that the color of the n'th cow bought doesn't depend on what the previously bought cow colors are. In your counterexample, there are 4 possible scenarios that are equally probable. You've counted two cases as a single case, which make 3 possible scenarios which are not all equally probable.

My personal experience is that it is like a pill that I believe is a placebo and so won't work. However I've known people who've been helped by religion. It can offer meaning, which is important to most for a sense of well-being. It can offer security in the face of uncertainty, and a feeling of being connected to something. Hope, etc. Answers to questions. A relief of some of the burden of personal responsibility (a moral guide so we don't have to decide everything for ourselves). I've also seen people use it as a false solution to problems, while the problems go ignored, sometimes leading to maybe a "crisis of faith". I've seen it become an overwhelming burden, a source of constant negativity, and a cause of fear and doubt. In summary, whether religion is your life, or a comfort, or a help or a crutch or a useless bag of turd, seems to be specific to the individual and likely a matter of preference.

You listed a few examples but you stopped before covering everything in the universe. Non-reversible processes shouldn't be cyclical. The Big Bang does not seem to be part of a cycle. Entropy of a system increases, and does not later return to a previous value. You should be able to detect a cyclical process by making forward and/or backward predictions and finding that the system will end up in the same state as a previous one.

Yes, because the first 4 would have to be of the form 4n, with a previous number including "nnnn", but there's no way to get that without splitting up a count of the digit n into two counts. Some patterns: 1) not including the first, all strings will end in alternating "11" and "21". Rough inductive proof: Suppose a number ends in "21". Then the next string will end in "?211", and the next will end in "?221". (Repeat induction step ad infinitum.) 2) Once the string starts with 13n (where n is not 3), it will repeat... 1113... 31m3... (for some possibly varying m) 13p1... for some p not 3. So the N'th number's first 2 and last 2 digits are a simple function of N. There seem to be bigger patterns too, at the starts and ends of the strings. I wonder if it's possible to figure out a function for the entire N'th string of numbers. Are there any possible numbers to start a similar sequence that form a "closed loop" and return to the same number? (I see that "22" is already mentioned.)

Here's a puzzle: When would you expect to see a 4?

No, that's not correct. Note that 49/99 is less than half. Why would finding out that one of the cows is brown make you think that less than half of the cows are brown? I'm not sure where the mistake is... perhaps step 4? Why is it multiplied by (n-1) instead of n? BTW the answer is much much easier to figure out!

No. The other 99 cows are not equally likely to be brown or blue.

The 100 cows specifically mentioned in the puzzle. Where they are is irrelevant. The only point in mentioning the barn is to justify the reasoning that you don't have any further info about the cows because they're hidden. This isn't a trick question. This is just a stats puzzle, not a lateral-thinking puzzle. The puzzle implies the existence of blue cows. That can be assumed.

False. It could also be straight lying, or delusion, or ignorance especially regarding how logical reasoning works, which can all occur independent of or in combination with religion.

Not my field either, and I'm in over my head now. I didn't mean for this to be a brain teaser or trick, just a quick stats puzzle. As mentioned by someone in a similar thread, these puzzles assume a Bayesian interpretation of probability, where it's possible to reason about an uncertain count of brown cows. So the expected count expresses probability, and can be fractional until enough information is known to make the count certain, in which case it would be integral. I meant for the expected proportion to represent the uncertain proportion, given only what information is in the puzzle. Sorry about the ambiguity!

No, that's not correct. The likelihood of the one brown cow being brown is 100% certain. I'll repeat my reply from the other thread because it's more appropriate here: Unless I worded it wrong, the expected proportion will be the mean of all equally probable possible proportions. The mean can be fractional. You've given the mode.

Unless I worded it wrong, the expected proportion will be the mean of all equally probable possible proportions. The mean can be fractional. You've given the mode.

Oh, I see. I might not have been confused if you'd used the word proper-velocity all along, even though it is not the simplest word. I don't think that it's a good trade-off to make things simpler at the expense of added ambiguity.

It is certainly an interesting way to put things, and an interesting way to explain the details of SR and maybe even show how or why they're consistent. Personally I would prefer that any conclusion made from a simple explanation would be equivalent to a conclusion made from the most difficult explanation of SR, in terms of wording. That way the simple understanding can't contradict a complete understanding. For example a complete understanding uses a precise definition of "speed", and a simple explanation shouldn't use the word with a different meaning just for the sake of simplicity. (To call it limitless speed in SR you're probably mixing reference frames, using one for time and another for distance.)

Models have a place in physics. I think it's important to separate a model's physically measurable predictions from any sort of metaphysical guess of what those measurements represent (the "underlying physical structure" that you mention). Then the value of any model begins and ends with its predictions, and not any extra implied meaning.

But ideally you'll want to interact with the rest of the universe, no??? If not, then what are you moving infinitely fast relative to? If you're moving a great distance, a great amount of time will pass. Sure, in your frame you can minimize both, but if you only care about your frame then you're not really moving at all. The time elsewhere in the universe is kind of important. This is what a lot of people describe as "time travel to the future", which is just you ageing little and everything around you ageing a lot. But the problem with a time-travel description is that you have to understand the meaning behind it before you can make sense of it without a lot of confusion. It's similar to the problem you face, which is that if you try to explain these things too simply, without explaining the prerequisite understanding, I'd think it more likely that people be mislead and confused than educated by it. I cringe to think of people arguing something like "you can exceed the speed of light, just by exploiting length contraction!" I think it's better to just understand the basics, which include "nothing can accelerate to faster than c", and then understand why and what it means (and your explanations probably fit there), and then only once one understands the basics would one get into how they can be exploited. If you understand the basics, then you understand the meaning of exploiting SR, and the caveats and the details.