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

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

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

"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

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

"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

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

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

"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.

"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