Daegs,

I tried an approximization at

http://www.wolframalpha.com/input/?i=sqr......52%29%29

for a 50% chance of a hit and it says the same as your guess. Way less than 52! in other words.

/Tomas

[quote]On 2009-09-04 01:54, Irfaan Kahan wrote:
I just shuffled a deck 7 times. Then I shuffled another deck seven times trying to copy my actions for the first deck as close as possible.

One of them is definitely more random than the other . . . but I'm not sure which.


Admittedly, I am not a math guy but but what does the statement "One of them is definitely more random than the other " even mean?

Also to produce a random result do we want to copy our actions from the first set of shuffles. I mean if you do faro shuffles then we are randomizing the decks/

Gil Scott

Tomas’ point is well taken. A question like “How many randomized decks do you need in order to have a 50% chance that two of them are in the same order?” is a useful way to get at the underlying question which is “In the history of playing cards, if every deck were shuffled into a truly random order (7 shuffles?), how likely is it that two of these decks have been in the same order?” It appears that it is extremely unlikely in face approaching zero probability.

My crazy scenario has every human who has ever lived be born shuffling decks 24 hrs a day for 75 years, each sufficiently shuffled to be random. If we give 15 seconds for 7 good riffle shuffles and assume the 96 billion value for the number of humans who have existed over the last 1 million years (check Wikipedia on this), you get this:

240 shuffles per hour (at 15 sec/shuffle) x 24 = 5760 shuffles/day = 2.104 x 10^6 (approx) shuffles/year = 1.578 x 10^8 shuffles in a lifetime for one person.

This leads to 1.52 x 10^19 randomized orders.

If we take Daeg’s and Tomas’ number of 1 x 10^34 as the number of decks needed for a 50% chance of two matching, we find that we’d need 10^15 TIMES as many decks. That’s 1,000,000,000,000,000 or 1 quadrillion. In other words we have 1/10^15 or
.000000000000001 or .0000000000001 percent of the needed decks to have a 50/50 chance of a match.

That’s how unlikely it is that any two really randomized decks have ever matched. Obviously cards haven’t been around for 1 million years and people can’t shuffle 24 hours a day for life, so the real probably of any two randomized decks ever having matched in the history of cards is far, far less than these number indicate.

Mike

I think part of my brain just exploded--or maybe imploded--my nose, eyes, and ears have begun to bleed...

A little philosophising here, but what's also worth pointing out that the randomness accorded to the decks is in reference to the fact that you have knowledge concerning the original starting position. So if a deck is in random order, and you give it a bunch of riffle shuffles, you might want to consider the deck after each of those shuffles as a random arrangement of cards, providing you knew nothing a priori.

That being said, I've seen a lot of guys shuffle cards a lot, and yet the 9 of diamonds still ends up on the bottom, and the 4 of clubs on top. Strange.

This post easily has enough mental stimulus to fry anyones brain. Michael's post could be used as an opener... You could then follow up with any material you wanted and the audience would still leave, a quivering mass of tapioca pudding.

Here's another way to blow your mind on the size of 52!

A 52 card Bicycle 808 deck weighs in at 3.1 oz. on my postal scale.
1 ounce = 28.3495231 grams so a deck weighs 87.88 grams or in standard metric that would be .08788 kg

A 52 card deck is about 1.6 cm thick or in standard metric that would be .016 meters.

The exact number of arrangements of a 52 card deck is 52! = 52*51*50*49…*3*2*1 which is about 8.06581752 × 10^67.

Thus if we had a stack consisting of every possible arrangement of cards is would contain 8.06581752 × 10^67 decks.

The weight of these decks would be 8.06581752 × 10^67 * .08788 kg which equals:

7.08824044 x 10^66 kg.

From: http://curious.astro.cornell.edu/question.php?number=342

“Now, the size of the observable universe is about 14 billion light years, and using the above value of density gives you a mass (dark and luminous matter) of about 3 x 10^55 g, which is roughly 25 billion galaxies the size of our galaxy, the Milky Way.”

So the estimated mass of the entire UNIVERSE (everything) is 3 x 10^55 grams which is 3 x 10^52 kg.

Thus the mass of the decks is about 2.36 x 10^14 TIMES as much as the mass of the entire universe. It would take more than 236 trillion (236 million million) universes to have enough mass to create the decks.

The distance from the bottom to the top of the stack of decks would be .016 m (thickness of a deck) x 8.06581752 x 10^67 = 1.2905 x 10^66 meters

From Wikipedia: The visible universe is thus a sphere with a diameter of about 28 billion parsecs (about 93 billion light-years-a different estimate from the earlier one?).

Since 1 light year = 9.4605284 × 10^15 meters, the visible universe is about 93 x 10^9 x 9.4605284 x 10^15 = 8.7983 x 10^26 meters.

But our deck stack is 1.2905 x 10^66. That would be across the visible universe 1.4668 x 10^39 times! That’s 1466.8 x 10^36 or 1266.8 trillion trillion trillion times across the visible UNIVERSE (not solor system or galaxy – UNIVERSE i.e. 93 billion light years).

The values for the mass and size of the universe get updated as new evidence comes in. The values used here are reasonable estimates.

It’s still impossible to fathom the size of 52!.

Mike

So that's a suitable measure to use when checking if you are in the right universe within the multiverse?

My head hurts.

My head feels fine , 'cause I don't care.

I had a piece of good fortune a few days ago. I randomly shuffled my deck and it ended up in Si Stebbins.

I've just got back from my long walk around the equator, and I'm not happy.

I took one droplet of water out of the pacific and it started raining, so I just piled sheets of paper up, and when they got near the sun the top one caught fire!

I think someones pulling my leg here.

There's no leg pulling. The math is easy to verify. Do 52! and multiply it by the mass of a deck. Then find online the mass of the known universe. The mass of the decks is unbelievably larger. The math is easy to validate.

Mike

What's scary to consider is that, as big numbers go, 52! isn't particularly.

Since the "set" of numbers is infinite, even 5200! "isn't"

Mike: Your thread is indeed a "Holy Terror"

Side note: We think about numbers as being infinite, but very few think of sentences in a given alphabetic language as being infinite, and yet, this is the case. Linguistically speaking, the AVERAGE vocabulary a person possesses is somewhere from 25,000-70,000 words. Yet from this finite set, infinite sentences are indeed possible. And, furthermore, a single sentence can be infinitely long - kind of like the Card Trick Game thread.

Vlad

Now we can get into a discussion of the possibility of making a book of say 400 pages consisting of random sequences of symbols from the keyboard including spaces. You can do the math on how many possible such books there could be: If there are say 256 possible such symbols for every slot on all the pages, you'd have 256^(400 x the number of symbols per page). This would be FAR larger than 52! and yet it would be a finite number. If you could make every possible such book the "library" would contain everything ever written as well as everything ever to be written in the future. It would also contain copies of each of these works that differ by one comma in one position or one letter here or two spaces there instead of 1 etc. So there would be tons of books that are nearly the same or are the same up to the last chapter but have totally different endings that make sense etc.

Check out Jorge Luis Borges story "The Library of Babel." Here's a wikipedia ref:

http://en.wikipedia.org/wiki/The_Library_of_Babel

Mike

Thanks for the link about the book - that's fascinating.

The Wiki page says:

"In any case, it is clear that a library containing all possible books, arranged at random, is equivalent (as a source of information) to a library containing zero books."

...but this can't be so if there exists the concept of "absolute truth". For example, "2 plus 2 equals 4" is a true statement, even in the presence of the statement "2 plus 2 does not equal 4".

So this leads to the question of how we know something to be true or not. Clearly, some of the maths (or logic) books could be discarded as false, and some kept as containing truth. Hence, there exists a theoretical library that contains the truth about everything (although we might not be able to identify all the books that belong in that library).

What do you think?

If there is "Truth" (with a capital T), then there would be a set of books "of the truth" in the library. This would be a cool addition to the Borges story since in the story there are purges of the library where groups search for bogus books and toss them into the abyss. Undoubtedly there would be groups of truth seekers whose mission it would be to purge the library of untrue books... These sorts exist even in our world now.

Mike

Well...I've just been pushed over to coin magic.

I just got back from the local library where I was searching for the book that is all letter X's. I told them I knew it existed, but after a frustrating time, I'm now banned from the library.

So much for that theory.