52, and other unimaginably large numbers

[Malintent]

It has been said that if you shuffle a deck of cards, the resulting arrangement of 52 cards will be unique... in that the odds are extremely high that never in the history of card-shuffling has that particular arrangement of cards ever existed. I thought that was pretty cool, so I looked more into it.

52 factorial, noted as "52!", represents the odds of there being a particular arrangement of 52 randomly ordered items. i.e. (52 x 51 x 50 x 49... x 2 x 1) = 52!

This is a large number. "How large?" is the focus of this.

52! is approximately equal to 8.066x10^67

The Universe is approximately 4.4x10^17 SECONDS old.

If you were to start arranging cards at a rate of 1 deck per second (you are a very fast shuffler) starting at the Big Bang, by now you will have not even scratched the surface of the number of possible arrangements of a deck of cards. This is how we know that is it incredibly unlikely that a deck of shuffled cards has ever been seen before.

So how long IS 52! seconds, to put it into perspective?

Try this experiment, if you have the time:

Pick a spot on Earth's equator.
Start a timer that counts the seconds to 52!
Stand in that spot for 1 BILLION YEARS
After 1 billion years, take a single step East
Wait another 1 billion years, and take another step...
After you have walked all of the way around the world, take 1 drop of water out of the pacific ocean and discard it (*poof*)
Repeat this process of taking 1 step every 1 billion years and removing a drop of water after each complete trip around the world.
Once you have completely drained the Pacific ocean, take a sheet of paper and place it on the ground.
Now, the ocean is magically refilled, so you can repeat all of the steps above, placing another piece of paper on top of the previous one after completely draining the pacific one drop at a time as you make it around the world again (1 step per 1 billion years).
Once you have stacked up enough sheets of paper to reach THE SUN, do you think you will have run out of time?

Nope. At this point, you are not quite 5% there. Repeat all of those steps from the very beginning approximately 250 more times, and you will then have lived long enough to have created every possible arrangement of cards, as long as you arranged them non-stop 1 time every second.

Large numbers are... really large.

 

[beero1000]

I see your  factorial function and raise you the  Ackermann function.

To get an idea of how fast that grows, A(3,3) = 61 and A(4,4) > ((52!)!)!

 

[fromderinside]

So I assume the POWERBALL people ran an expectation of about 50 of million betters every week hit the Powerball jackpot, on average, about once every 5 weeks. or at a rate of 1 in 300,000,000

Powerball odds: http://www.smartluck.com/free-lottery-tips/powerball-569-pb.htm

Which I think is why their number calculated from the empirical rate of winning at a rate of once every five weeks comes to about i in 292 million as opposed to the weekly expectation of a winner calculation results in 69!/(64!+5!) + 1!/26!/(1!+25!) or 1 in 13,482,621,560

So the expected rate of a winner seems, empirically, to be about one fifth of the calculated odds or about the number entered in the article, 292 million.

I got this from my study of the probability of association effects by adding a rings of Saturn variable to a correlation.

... so I agree with beero1000 starting from raw numbers doesn't take into account actual rates of achieving a second hand with the same arrangement which needs inclusion for accounting for number of shufflers and rates of shuffling

What if we presume 100 million folders folding at a rate of one hundred hands per hour. Seems to me the probability of a second hand identical to the first hand becomes a lot more likely. I mean if there are 100 million original hands don't you think one of them would achieve a second identical hand a lot faster than the naked one arrangement of one original hand by an original shuffler?

 

[PyramidHead]

I still don't get the title. Shouldn't it be "52!" instead of "52"?

 

[beero1000]

So I assume the POWERBALL people ran an expectation of about 50 of million betters every week hit the Powerball jackpot, on average, about once every 5 weeks. or at a rate of 1 in 300,000,000

Powerball odds: http://www.smartluck.com/free-lottery-tips/powerball-569-pb.htm

Which I think is why their number calculated from the empirical rate of winning at a rate of once every five weeks comes to about i in 292 million as opposed to the weekly expectation of a winner calculation results in 69!/(64!+5!) + 1!/26!/(1!+25!) or 1 in 13,482,621,560

So the expected rate of a winner seems, empirically, to be about one fifth of the calculated odds or about the number entered in the article, 292 million.

The odds for the Powerball jackpot for a single play are \( 1 \text{ in } C(69,5) \cdot 26 = 1 \text{ in } 292,201,338\). If there are 50,000,000 plays per week, the probability of no one winning is \( (1 - 1/292,201,338)^{50,000,000} \approx 84%\), meaning that the expectation is about 1 winner every 6.35 weeks. To get a winner every 5 weeks, you'd need about 65.2 million plays per week. (This is ignoring the fact that the number of plays is definitely dependent on the amount of time since the last winner though, but you'd need to measure the distribution to get a better stat...)

I got this from my study of the probability of association effects by adding a rings of Saturn variable to a correlation.

... so I agree with beero1000 starting from raw numbers doesn't take into account actual rates of achieving a second hand with the same arrangement which needs inclusion for accounting for number of shufflers and rates of shuffling

What if we presume 100 million folders folding at a rate of one hundred hands per hour. Seems to me the probability of a second hand identical to the first hand becomes a lot more likely. I mean if there are 100 million original hands don't you think one of them would achieve a second identical hand a lot faster than the naked one arrangement of one original hand by an original shuffler?

Did I say that?

I still don't get the title. Shouldn't it be "52!" instead of "52"?

Presumably.

 

[fast]

8.066x10^67

4.4x10^17

I have a few questions, easy questions--just to make sure I'm not mistaken.

Does 8.066x10^67 fall between 10^67 and 10^68?
Does 4.4x10^17 fall between 10^17 and 10^18?

What does 8.066x10^3 and 4.4x10^3 look like long hand? I presume they both fall between 10^3 and 10^4. 8066 and 4400?

 

[Juma]

8.066x10^67

4.4x10^17

I have a few questions, easy questions--just to make sure I'm not mistaken.

Does 8.066x10^67 fall between 10^67 and 10^68?
Does 4.4x10^17 fall between 10^17 and 10^18?

What does 8.066x10^3 and 4.4x10^3 look like long hand? I presume they both fall between 10^3 and 10^4. 8066 and 4400?

Yes, Yes, 8066, 4400, yes, yes.
10x10^n = 1x10^(n+1)
Nuff said.
How come you dont know this?

 

[J842P]

I have a few questions, easy questions--just to make sure I'm not mistaken.

Does 8.066x10^67 fall between 10^67 and 10^68?
Does 4.4x10^17 fall between 10^17 and 10^18?

What does 8.066x10^3 and 4.4x10^3 look like long hand? I presume they both fall between 10^3 and 10^4. 8066 and 4400?

Yes, Yes, 8066, 4400, yes, yes.
10x10^n = 1x10^(n+1)
Nuff said.
How come you dont know this?

The quality of mathematics and science education at the secondary/primary level is so fractured in the US that large swaths of the population are effectively innumerate.

However, I think fast is just rusty, and probably hasn't had to use this stuff since he learned it a while back. Indeed, he *did* know it, he just wanted to make sure his recollection was accurate.

 

[beero1000]

I have a few questions, easy questions--just to make sure I'm not mistaken.

Does 8.066x10^67 fall between 10^67 and 10^68?
Does 4.4x10^17 fall between 10^17 and 10^18?

What does 8.066x10^3 and 4.4x10^3 look like long hand? I presume they both fall between 10^3 and 10^4. 8066 and 4400?

Yes, Yes, 8066, 4400, yes, yes.
10x10^n = 1x10^(n+1)
Nuff said.
How come you dont know this?

That's a good way to make sure people don't ask questions when they aren't sure about something. Is that something you want to happen?

- - - Updated - - -

8.066x10^67

4.4x10^17

I have a few questions, easy questions--just to make sure I'm not mistaken.

Does 8.066x10^67 fall between 10^67 and 10^68?
Does 4.4x10^17 fall between 10^17 and 10^18?

What does 8.066x10^3 and 4.4x10^3 look like long hand? I presume they both fall between 10^3 and 10^4. 8066 and 4400?

All good.

 

[Malintent]

I still don't get the title. Shouldn't it be "52!" instead of "52"?

Yes, that would have been more accurate but less "fantastic" sounding.. hehe.

"deep time" / "large numbers" are interesting to me because it speaks to human incredulity (to evolution, age of universe, etc...)

 

[Juma]

Yes, Yes, 8066, 4400, yes, yes.
10x10^n = 1x10^(n+1)
Nuff said.
How come you dont know this?

The quality of mathematics and science education at the secondary/primary level is so fractured in the US that large swaths of the population are effectively innumerate.

However, I think fast is just rusty, and probably hasn't had to use this stuff since he learned it a while back. Indeed, he *did* know it, he just wanted to make sure his recollection was accurate.

If you once understood potenses you dont forget that. I was honestly curious how one can go through education without learning it.

 

[Malintent]

8.066x10^67

4.4x10^17

I have a few questions, easy questions--just to make sure I'm not mistaken.

Does 8.066x10^67 fall between 10^67 and 10^68?
Does 4.4x10^17 fall between 10^17 and 10^18?

What does 8.066x10^3 and 4.4x10^3 look like long hand? I presume they both fall between 10^3 and 10^4. 8066 and 4400?

This is what the number of seconds since the Big Bang looks like:

440,000,000,000,000,000

Now, add 50 more zeros to the end... that's what the number of possible 52 card decks looks like.

 

[fast]

I've always liked math; it's just that compared to you guys, it's not my strong suit.

Anyhow, in spirit of contributing unimaginably large numbers, I thought I'd throw one in the ring for fun:
Hey, might can't do the math, but that don't mean I can't turn it into a word problem

P=1.616 x 10^-35 meters
U=8.798 x 10^23 meters

How big is the universe measured in Planck length?

To illustrate, imagine a very long road; how long? See U above.

Now, imagine placing some very small objects end to end following the entire length of the road. How small? see P above.

Want to know how many objects are on the road? Yeah, me too. But, for a truly unimaginable number, imagine the volume of the largest estimated universe size (less than infinity) measured by how many P's it could theoretically hold unimpeded by laws of physics. Then, treat each of those as done in the factorial OP example. That takes us farther than what can be imagined, and no, you can't add one and prove a point because I still can't imagine two less than adding yet another.

 

[untermensche]

This is what the number of seconds since the Big Bang looks like:

440,000,000,000,000,000

Now, add 50 more zeros to the end... that's what the number of possible 52 card decks looks like.

If two decks of cards are shuffled randomly and vigorously and both end up in the same order what are the chances of that?

Answer: 100%, It happened.

 

[Bomb#20]

It has been said that if you shuffle a deck of cards, the resulting arrangement of 52 cards will be unique... in that the odds are extremely high that never in the history of card-shuffling has that particular arrangement of cards ever existed. I thought that was pretty cool, so I looked more into it.

52 factorial, noted as "52!", represents the odds of there being a particular arrangement of 52 randomly ordered items. i.e. (52 x 51 x 50 x 49... x 2 x 1) = 52!

This is a large number. "How large?" is the focus of this.

52! is approximately equal to 8.066x10^67

This assumes if you shuffle a deck you're equally likely to get one arrangement as any other. That isn't the case. Some arrangements are vastly more likely than others, especially when you shuffle a brand new deck for the first time. Assuming you do a riffle shuffle, there are probably only a few million plausible outcomes, because which half of the deck the next card comes from is strongly correlated with which half the last card came from. Since millions of packs of cards are sold, the chances are pretty good that somebody else got the same result as you. (Of course, if you sensibly shuffle it several times you can drive the chance of that down to never in the history of card-shuffling. )

 

[Malintent]

This is what the number of seconds since the Big Bang looks like:

440,000,000,000,000,000

Now, add 50 more zeros to the end... that's what the number of possible 52 card decks looks like.

If two decks of cards are shuffled randomly and vigorously and both end up in the same order what are the chances of that?

Answer: 100%, It happened.

Stated in the past tense, that is correct. Stated in the future tense, it would be, "The chances of a second deck matching the first is 1/52!"

There are no "chances" of anything happening in the past, unless there is a component of quantum mechanics we as of yet do not understand that creates a future potential for changes to past events... in which case the words "event", and "happened" take on a whole new meaning as to the temporary status of "events".

 

[Malintent]

It has been said that if you shuffle a deck of cards, the resulting arrangement of 52 cards will be unique... in that the odds are extremely high that never in the history of card-shuffling has that particular arrangement of cards ever existed. I thought that was pretty cool, so I looked more into it.

52 factorial, noted as "52!", represents the odds of there being a particular arrangement of 52 randomly ordered items. i.e. (52 x 51 x 50 x 49... x 2 x 1) = 52!

This is a large number. "How large?" is the focus of this.

52! is approximately equal to 8.066x10^67

This assumes if you shuffle a deck you're equally likely to get one arrangement as any other. That isn't the case. Some arrangements are vastly more likely than others, especially when you shuffle a brand new deck for the first time. Assuming you do a riffle shuffle, there are probably only a few million plausible outcomes, because which half of the deck the next card comes from is strongly correlated with which half the last card came from. Since millions of packs of cards are sold, the chances are pretty good that somebody else got the same result as you. (Of course, if you sensibly shuffle it several times you can drive the chance of that down to never in the history of card-shuffling. )

For the purposes of this thread (and for being able to model statistics, probability, and other math) we are believing in "pure randomness". I am in agreement with you that, in the real (macro) world, there is no such thing as "random".... for the reasons you state, as an example. In the real world, there is no pure randomness... only "indeterminism". In the world of mathematical models, for fairness, we assume true randomness... like how when we model Newtonian physics, we ignore air resistance for understanding, and then in engineering we correct for it.

 

[Peez]

If two decks of cards are shuffled randomly and vigorously and both end up in the same order what are the chances of that?

Answer: 100%, It happened.

Stated in the past tense, that is correct. Stated in the future tense, it would be, "The chances of a second deck matching the first is 1/52!"

There are no "chances" of anything happening in the past, unless there is a component of quantum mechanics we as of yet do not understand that creates a future potential for changes to past events... in which case the words "event", and "happened" take on a whole new meaning as to the temporary status of "events".

One of the major problems encountered in time travel is not that of becoming your own father or mother. There is no problem in becoming your own father or mother that a broad-minded and well-adjusted family can't cope with. There is no problem with changing the course of history—the course of history does not change because it all fits together like a jigsaw. All the important changes have happened before the things they were supposed to change and it all sorts itself out in the end.

The major problem is simply one of grammar, and the main work to consult in this matter is Dr. Dan Streetmentioner's Time Traveler's Handbook of 1001 Tense Formations. It will tell you, for instance, how to describe something that was about to happen to you in the past before you avoided it by time-jumping forward two days in order to avoid it. The event will be descibed differently according to whether you are talking about it from the standpoint of your own natural time, from a time in the further future, or a time in the further past and is futher complicated by the possibility of conducting conversations while you are actually traveling from one time to another with the intention of becoming your own mother or father.

Most readers get as far as the Future Semiconditionally Modified Subinverted Plagal Past Subjunctive Intentional before giving up; and in fact in later aditions of the book all pages beyond this point have been left blank to save on printing costs.

The Hitchhiker's Guide to the Galaxy skips lightly over this tangle of academic abstraction, pausing only to note that the term "Future Perfect" has been abandoned since it was discovered not to be.

From The Restaurant at the End of the Universe

 

[CJW]

Nope. At this point, you are not quite 5% there. Repeat all of those steps from the very beginning approximately 250 more times, and you will then have lived long enough to have created every possible arrangement of cards, as long as you arranged them non-stop 1 time every second.

Large numbers are... really large.

Are you missing a decimal? Should that be: "not quite 0.5%" ?

I wonder if there is a way to estimate how many standard 52 card decks have been shuffled since they were first introduced.

 

[J842P]

The quality of mathematics and science education at the secondary/primary level is so fractured in the US that large swaths of the population are effectively innumerate.

However, I think fast is just rusty, and probably hasn't had to use this stuff since he learned it a while back. Indeed, he *did* know it, he just wanted to make sure his recollection was accurate.

If you once understood potenses you dont forget that. I was honestly curious how one can go through education without learning it.

Sure you could forget it. Most people could go through their entire lives without using exponents ever again after they pass some exam in secondary school. Regardless, it is clear he did learn it. He just wasn't sure if his recollection was accurate.