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Billions & billions
Bear with me. Helped a schoolkid with a quote originally from Carl Sagan:
"A handful of sand contains about 10,000 grains, more than the total number of stars we can see with the naked eye on a clear night. But the number of stars we can see is only the tiniest fraction of the number of stars that are. What we see at night is the merest smattering of the nearest stars with a few more distant, bright stars thrown in for good measure. Meanwhile, the cosmos is rich beyond measure. The total number of stars in the universe is larger than all the grains of sand on all the beaches of the planet earth." Kind of an echo of a biblical verse, Genesis 22:17: "That in blessing I will bless thee, and in multiplying I will multiply thy seed as the stars of the heaven, and as the sand which is upon the sea shore;" A 2003 news story was more exact, pinning the number of stars in the known universe as 70 sextillion. http://www.cnn.com/2003/TECH/space/07/22/stars.survey/ That's a lot of stars. 7x10^22 Made me think about how any individual can be unique in the universe--or do we have doppelgangers out there on far-flung star systems? Our uniqueness is in part due to our DNA sequences, so a question might be how many combinations are there of the human genome? Or as a illustrative start, how many different ways are there to arrange a deck of 52 playing cards? This is the surprising result (to me anyway): Google informs me that 52 factorial (the number of ways 52 unique cards can be arranged in a deck) is the staggering sum of 8.06581752 × 10^67. Therefore, the number of different shuffles of a 52-card deck vastly dwarfs the number of stars in the observable universe! So much so that if every star had ten planets, each planet with 10 billion inhabitants who have been shuffling cards continually for the last 15 billion years (age of the universe), each producing a new (random) shuffle every second, the odds are vanishingly small that the same shuffle would have been produced twice to date! Specifically about one chance in 10^16 in this extreme example. One chance in ten million billion. Suddenly the universe doesn't seem big enough to encompass the variation in a simple deck of cards? Joe |
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Billions & billions
Ï "Joe Knapp" Ýãñáøå óôï ìÞíõìá .com... Bear with me. Helped a schoolkid with a quote originally from Carl Sagan: [snip] Suddenly the universe doesn't seem big enough to encompass the variation in a simple deck of cards? Wait till you find out how many different combinations are possible in our DNA, genewise. :*))) Joe -- Ioannis Galidakis http://users.forthnet.gr/ath/jgal/ ------------------------------------------ Eventually, _everything_ is understandable |
#3
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Billions & billions
Joe Knapp wrote:
Or as a illustrative start, how many different ways are there to arrange a deck of 52 playing cards? This is the surprising result (to me anyway): Google informs me that 52 factorial (the number of ways 52 unique cards can be arranged in a deck) is the staggering sum of 8.06581752 × 10^67. Therefore, the number of different shuffles of a 52-card deck vastly dwarfs the number of stars in the observable universe! So much so that if every star had ten planets, each planet with 10 billion inhabitants who have been shuffling cards continually for the last 15 billion years (age of the universe), each producing a new (random) shuffle every second, the odds are vanishingly small that the same shuffle would have been produced twice to date! Specifically about one chance in 10^16 in this extreme example. One chance in ten million billion. No, that isn't right. This problem is analogous to the so-called "birthday paradox." Given a room with N people, presumably with their birthdays randomly (and uniformly, let's assume) distributed, what is the probability p that at least two of them share the same birthday? For what value of N would you expect p to be about 1/2? N = 100? N = 50? Those who haven't seen this problem before are generally surprised that p is most nearly 1/2 when N = 23. The statistics can be reasoned out as follows. When N = 1, p is very obviously 0. When N = 2, the probability that there is *no* match is just the probability that the second person's birthday is not the same as the first: 364/365. So the probability of a match is just 1 - 364/365 = 1/365 When N = 3, the probability that there is *no* match is the probability that the first two people didn't match (which we saw was 364/365), *times* the probability that the third person doesn't match either of the first two, which is 363/365. That is, (364/365)(363/365) = 132,132/133,225, and the probability of a match is 132,132 1,093 1 - ------- = ------- 133,225 133,225 In general, for any N, the probability that there is *no* match is equal to 365 364 363 365-N+1 --- x --- x --- x ... x ------- 365 365 365 365 and the probability of a match is then 1 minus that product. It so happens that for N = 23 that p is very close to 1/2. We can extend this problem to the one you discuss. Instead of having 365 days in a year, we have 52! = 8 x 10^67 (approximately) different shuffles of a full deck of cards. And instead of people in a room, we have different samples. The number of samples you consider is (7 x 10^22 stars)(10 planets/star)(10^10 inhabitants/planet) (1 sample/inhabitant/second)(3 x 10^7 seconds/year)(1.5 x 10^10 years) which is about N = 3 x 10^51 samples. The probability of no match is then 52! 52!-1 52!-2 52!-3 52!-N+1 --- x ----- x ----- x ----- x ... x ------- 52! 52! 52! 52! 52! Since N is so small in comparison with 52! (about 1 part in 10^16, as you note), we might expect this product to be close to 1, but consider that this product is actually *smaller* than (but very close to) 52!-(N/2) 52!-(N/2) 52!-(N/2) --------- x --------- x --------- x ... 52! 52! 52! with the same term multiplied by itself N times, in the same way that 365 x 364 x 363 x 362 x 361 x 360 x 359 362^7 because 363 x 361 362^2 364 x 360 362^2 and 365 x 359 362^2 But that product can also be written as +- -+ N | N/2 | | 1 - ------- | | 52! | +- -+ which has the approximation e^[-(N^2)/(2 x 52!)] The part in the brackets is approximately 9 x 10^102 divided by twice 8 x 10^67, or about 5 x 10^34. So the probability that there is no match is actually vanishingly small: about 1 chance in 50 thousand million million million million million. It is overwhelmingly likely that somewhere amongst the 3 x 10^51 samples that there is a match--in fact, that there are lots and lots of matches. When does p = 1/2 approximately? For N such that N^2/(2 x 52!) is about 0.7--the natural logarithm of 2. That gives us N^2 = 1.4 x 52! = 10^68, or N = 10^34 For the birthday paradox, we have N^2 = 1.4 x 365 = 511, or N = 22.6, close enough Brian Tung The Astronomy Corner at http://astro.isi.edu/ Unofficial C5+ Home Page at http://astro.isi.edu/c5plus/ The PleiadAtlas Home Page at http://astro.isi.edu/pleiadatlas/ My Own Personal FAQ (SAA) at http://astro.isi.edu/reference/faq.txt |
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Billions & billions
"Brian Tung" wrote in message ... Joe Knapp wrote: Or as a illustrative start, how many different ways are there to arrange a deck of 52 playing cards? This is the surprising result (to me anyway): Google informs me that 52 factorial (the number of ways 52 unique cards can be arranged in a deck) is the staggering sum of 8.06581752 × 10^67. Therefore, the number of different shuffles of a 52-card deck vastly dwarfs the number of stars in the observable universe! So much so that if every star had ten planets, each planet with 10 billion inhabitants who have been shuffling cards continually for the last 15 billion years (age of the universe), each producing a new (random) shuffle every second, the odds are vanishingly small that the same shuffle would have been produced twice to date! Specifically about one chance in 10^16 in this extreme example. One chance in ten million billion. No, that isn't right. This problem is analogous to the so-called "birthday paradox." Given a room with N people, presumably with their birthdays randomly (and uniformly, let's assume) distributed, what is the probability p that at least two of them share the same birthday? You are right--thanks for that, it's a keeper. So I will modify the example to: * shuffle a deck of cards * the chance of that shuffle being replicated anywhere in the known universe, given the extreme conditions above, are one chance in 10 million billion And even with the birthday paradox situation you outline, you calculate that 10^34 shuffles are needed to produce a 50% chance of an unconstrained repeat. Which works out to 10^11 (100 billion) shuffles per star in the latest inventory. Joe |
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Billions & billions
Joe Knapp:
Bear with me. Helped a schoolkid with a quote originally from Carl Sagan: "A handful of sand contains about 10,000 grains, more than the total number of stars we can see with the naked eye on a clear night. But the number of stars we can see is only the tiniest fraction of the number of stars that are. What we see at night is the merest smattering of the nearest stars with a few more distant, bright stars thrown in for good measure. Meanwhile, the cosmos is rich beyond measure. The total number of stars in the universe is larger than all the grains of sand on all the beaches of the planet earth." Kind of an echo of a biblical verse, Genesis 22:17: "That in blessing I will bless thee, and in multiplying I will multiply thy seed as the stars of the heaven, and as the sand which is upon the sea shore;" A 2003 news story was more exact, pinning the number of stars in the known universe as 70 sextillion. http://www.cnn.com/2003/TECH/space/07/22/stars.survey/ That's a lot of stars. 7x10^22 Made me think about how any individual can be unique in the universe--or do we have doppelgangers out there on far-flung star systems? Our uniqueness is in part due to our DNA sequences, so a question might be how many combinations are there of the human genome? Or as a illustrative start, how many different ways are there to arrange a deck of 52 playing cards? This is the surprising result (to me anyway): Google informs me that 52 factorial (the number of ways 52 unique cards can be arranged in a deck) is the staggering sum of 8.06581752 × 10^67. Therefore, the number of different shuffles of a 52-card deck vastly dwarfs the number of stars in the observable universe! So much so that if every star had ten planets, each planet with 10 billion inhabitants who have been shuffling cards continually for the last 15 billion years (age of the universe), each producing a new (random) shuffle every second, the odds are vanishingly small that the same shuffle would have been produced twice to date! Specifically about one chance in 10^16 in this extreme example. One chance in ten million billion. Suddenly the universe doesn't seem big enough to encompass the variation in a simple deck of cards? Impressive numbers -- for those who are impressed by numbers. But, as far as we know, the sentient population of the Universe is about six billion beings. I believe that the _Hitchhiker's Guide to the Galaxy_ suggests that the population of the Universe is zero, because the number of places where life is known to exist, divided by the number of places where life could exist, is so close to zero as to be mathematically indistinguishable from zero. (Actually, the HHGG took some license here, as in other areas, and I believe that it said that the number of places where life could exist was infinite, and that any finite number divided by infinity equals zero.) Close enough for my purposes, and easier to grasp than 8.06581752 X 10^67. The notion that we could be alone in the Universe is neither surprising nor disturbing to me. If you want to amuse your non-mathematical friends with factorials, consider the simple 17-student question. A teacher has a small class of only 17 students, and there are 17 chairs for them to sit in. She wants to try every possible seating arrangement. It takes her half an hour to work out each arrangement. How long will it take her to work out every possible arrangement if she works 24 hours per day until completion? Davoud Hint: If she started during Planck Time, she wouldn't be finished yet. -- usenet *at* davidillig dawt com |
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Billions & billions
"Davoud" wrote The notion that we could be alone in the Universe is neither surprising nor disturbing to me. Nice to know. Must be why you use a Mac? Joe |
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Billions & billions
Davoud:
The notion that we could be alone in the Universe is neither surprising nor disturbing to me. Joe Knapp: Nice to know. Must be why you use a Mac? Haven't you heard? *Nobody* uses a Mac. Please see http://www.davidillig.com/aas/who.shtml for proof. Davoud -- usenet *at* davidillig dawt com |
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Billions & billions
There are numbers, and there are 'significant numbers'. (not to mention
significant relationships vs nonsignificant). Jerry Joe Knapp wrote: Bear with me. Helped a schoolkid with a quote originally from Carl Sagan: "A handful of sand contains about 10,000 grains, more than the total number of stars we can see with the naked eye on a clear night. But the number of stars we can see is only the tiniest fraction of the number of stars that are. What we see at night is the merest smattering of the nearest stars with a few more distant, bright stars thrown in for good measure. Meanwhile, the cosmos is rich beyond measure. The total number of stars in the universe is larger than all the grains of sand on all the beaches of the planet earth." Kind of an echo of a biblical verse, Genesis 22:17: "That in blessing I will bless thee, and in multiplying I will multiply thy seed as the stars of the heaven, and as the sand which is upon the sea shore;" A 2003 news story was more exact, pinning the number of stars in the known universe as 70 sextillion. http://www.cnn.com/2003/TECH/space/07/22/stars.survey/ That's a lot of stars. 7x10^22 Made me think about how any individual can be unique in the universe--or do we have doppelgangers out there on far-flung star systems? Our uniqueness is in part due to our DNA sequences, so a question might be how many combinations are there of the human genome? Or as a illustrative start, how many different ways are there to arrange a deck of 52 playing cards? This is the surprising result (to me anyway): Google informs me that 52 factorial (the number of ways 52 unique cards can be arranged in a deck) is the staggering sum of 8.06581752 × 10^67. Therefore, the number of different shuffles of a 52-card deck vastly dwarfs the number of stars in the observable universe! So much so that if every star had ten planets, each planet with 10 billion inhabitants who have been shuffling cards continually for the last 15 billion years (age of the universe), each producing a new (random) shuffle every second, the odds are vanishingly small that the same shuffle would have been produced twice to date! Specifically about one chance in 10^16 in this extreme example. One chance in ten million billion. Suddenly the universe doesn't seem big enough to encompass the variation in a simple deck of cards? Joe |
#9
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Billions & billions
"Joe Knapp" wrote in message y.com...
Bear with me. Helped a schoolkid with a quote originally from Carl Sagan: "A handful of sand contains about 10,000 grains, more than the total number of stars we can see with the naked eye on a clear night. But the number of stars we can see is only the tiniest fraction of the number of stars that are. What we see at night is the merest smattering of the nearest stars with a few more distant, bright stars thrown in for good measure. Meanwhile, the cosmos is rich beyond measure. The total number of stars in the universe is larger than all the grains of sand on all the beaches of the planet earth." Kind of an echo of a biblical verse, Genesis 22:17: "That in blessing I will bless thee, and in multiplying I will multiply thy seed as the stars of the heaven, and as the sand which is upon the sea shore;" A 2003 news story was more exact, pinning the number of stars in the known universe as 70 sextillion. http://www.cnn.com/2003/TECH/space/07/22/stars.survey/ That's a lot of stars. 7x10^22 Just putting that expression in a spreadsheet: 70,000,000,000,000,000,000,000 You can do some simple mathematics to find out how long it would take you to view each star at the rate of one viewing per second (that's 3,600 an hour) 19,444,444,444,444,400,000 - hours 810,185,185,185,185,000 - days 2,219,685,438,863,520 - years 29,595,805,851,514 - lifetimes (at 75 years per life) 34,682,585 - post dinosaur periods (that's the 64million years between dinosaur extinction until now) 482,540 - Periods from the creation of the earth (4,600,000,000 years ago) until now. Remember, you're seeing one star per second! (and i'm hoping i got the above right!) Infact, you'd only get to see a tiny proportion of those stars even it it was possible. Over time stars grow old and die, the longest lived stars (ones so small they 'burn' their fuel slowly) can exist as stars (from memory) for tens of billions of years. Our sun has about a 10 billion years lifespan (it's about half way through). New stars come into being, but rely on a finite amount of suitable materials in the universe, so each generation is less in number (presumably, because suitable materials are forever altered by previous star generations). At some point in the future, given current models of science, there would be no more stars. Just empty blackness, the occasional clump of matter not yet fallen into a black hole. Even black holes are set to vanish over time. Perhaps, at the end of the time period shown above, there would be nothingness, with nothing to be relative to things such as distance have no meaning, even time would have no real meaning. But that's speculation, really. The more immediate threats to us, in terms of facing our own nothingness, are in rough guess order of 'threat' 1. Something we do to wreck earth for us 2. A big asteroid collision 3. A really huge solar flare or other solar event doing some irreperable damage 4. A 'nearby' supernova radiating the place for light years around 5. The sun becoming a red giant 1 - seems a constant concern, 2 - is regular enough to be worth worrying about (about 1% chance per 1000 years ive read), 3 - no evidence that our sun gets this nasty!, 4 - don't know if we're near enough to any likely candidates, 5 - virtually guaranteed, but we do have some time! I can't even begin to imagine (at least with any sense of realism) what humankind, if it exists, would be like in a mere 10,000,000 years! Made me think about how any individual can be unique in the universe--or do we have doppelgangers out there on far-flung star systems? Our uniqueness is in part due to our DNA sequences, so a question might be how many combinations are there of the human genome? As indicated elsethread, the chances become even more astronomical (!) with dna. It seems both very difficult to think that the exact sequence of events that lead to intelligent (well, us lot) life here on earth has happened elsewhere, and at once very difficult to think it simply hasn't happened anywhere among those stars. I would believe that there is life out there, but that the kind of life that becomes interested in space and space exploration is a vanishingly small proportion, making it possible we're the only ones. We may be the only 'life' bearing planet. With our current rate of space exploration i wouldn't hold bet on bing able to answer the question 'is/was there life in our solar system other than on earth?' in the next few decades, maybe more. |
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Billions & billions
"gswork" wrote 7x10^22 Just putting that expression in a spreadsheet: 70,000,000,000,000,000,000,000 You can do some simple mathematics to find out how long it would take you to view each star at the rate of one viewing per second (that's 3,600 an hour) 19,444,444,444,444,400,000 - hours 810,185,185,185,185,000 - days 2,219,685,438,863,520 - years 29,595,805,851,514 - lifetimes (at 75 years per life) 34,682,585 - post dinosaur periods (that's the 64million years between dinosaur extinction until now) 482,540 - Periods from the creation of the earth (4,600,000,000 years ago) until now. Remember, you're seeing one star per second! (and i'm hoping i got the above right!) A lot of stars no doubt. But still far fewer than the number of different shuffles of a 52-card deck. In fact, a 23-card deck would do it. So take one complete suit and ace through ten of another. Our sun could be assigned to one shuffle of that, as could every other star. Infact, you'd only get to see a tiny proportion of those stars even it it was possible. Over time stars grow old and die, the longest lived stars (ones so small they 'burn' their fuel slowly) can exist as stars (from memory) for tens of billions of years. Our sun has about a 10 billion years lifespan (it's about half way through). New stars come into being, but rely on a finite amount of suitable materials in the universe, so each generation is less in number (presumably, because suitable materials are forever altered by previous star generations). At some point in the future, given current models of science, there would be no more stars. Just empty blackness, the occasional clump of matter not yet fallen into a black hole. Even black holes are set to vanish over time. Perhaps, at the end of the time period shown above, there would be nothingness, with nothing to be relative to things such as distance have no meaning, even time would have no real meaning. But that's speculation, really. On the black holes vanishing, that's Stephen Hawking's result that they "evaporate," right? I recently read his book Universe in a Nutshell & he mentions that the "temperature" of a black hole is some incredibly small figure like 1 million billionth of a degree Kelvin or something like that. So they don't disappear within the expected lifetime of the universe? The more immediate threats to us, in terms of facing our own nothingness, are in rough guess order of 'threat' 1. Something we do to wreck earth for us 2. A big asteroid collision 3. A really huge solar flare or other solar event doing some irreperable damage 4. A 'nearby' supernova radiating the place for light years around 5. The sun becoming a red giant 1 - seems a constant concern, 2 - is regular enough to be worth worrying about (about 1% chance per 1000 years ive read), 3 - no evidence that our sun gets this nasty!, 4 - don't know if we're near enough to any likely candidates, 5 - virtually guaranteed, but we do have some time! I can't even begin to imagine (at least with any sense of realism) what humankind, if it exists, would be like in a mere 10,000,000 years! The consolation might be that all those calamities are probably sufficiently remote (including the asteroid strike) that we don't have to worry about them in this year's fiscal budget. The chance of a significant asteroid strike in the next 1000 years is miniscule, and imagine what solar system travel and technology will be 1000 years hence. It seems both very difficult to think that the exact sequence of events that lead to intelligent (well, us lot) life here on earth has happened elsewhere, and at once very difficult to think it simply hasn't happened anywhere among those stars. I would believe that there is life out there, but that the kind of life that becomes interested in space and space exploration is a vanishingly small proportion, making it possible we're the only ones. We may be the only 'life' bearing planet. FWIW. I also believe that there is life throughout the universe, and also agree with you about interstellar travel. If the speed limit is 'c' then that explains Fermi's paradox--interstellar travel is impractical everywhere and always will be. Or interstellar communication for that matter--a twenty-minute roundtrip delay to Mars, a mere stone's throw away, is bad enough for communication. But to believe we are alone, the chosen people of the universe, smacks of religious claptrap to me, and fundamentally unscientific, akin to believing the Earth is at the center of the universe. Nature abhors discontinuities, namely the abrupt increase in entropy when highly organized stellar radiation strikes the surface of a rocky planet and is rapidly degraded into heat. Life is a natural reaction to this cliff of negative entropy, smoothing it out. Plants literally rise up towards the light in branching patterns like a stream flowing over a terrestrial cliff, eroding it and smoothing it out. It's clear that there are certain necessary conditions for life to arise (such as a certain temperature range, liquid water or perhaps other high-energy phenomena like lightning), but there are at least 7x10^22 solar systems to choose from. Life's not an abstract random fluke, like a certain shuffle of the cards, but more like a stacked deck, ordered by the laws of thermodynamics. Joe |
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