Has EVERYTHING happened in poker?

[CaptainNurple]

(mods: if you think this fits better in the poker theory forum, please move it there)

Perhaps this has been discussed before, if so feel free to post links, but this seems like a fun problem to me:

1) Given a 10-handed Hold 'Em table (or whatever constraints you want), how large is the complete set of every possible scenario?
2) Based on this number, estimate the likelyhood that "everything" has happened in Hold 'Em in the history of the game. Or, alternately, estimate the average length of time required for "everthing" to happen, stating your assumptions (e.g. average hands per hour, one table versus all the tables in the world-how many?, etc)

Another way of thinking about this problem is to pick any possible scenario, (e.g. pocket 2's versus pocket 7's, flop of A27, turn is a 2, river is a 7--including all suits and their distribution) then wonder if it's ever occurred. Then extend that to include every possible scenario.

What's interesting to me about this is that you can easily calculate the probability of any one weird/unlikely event, and for isolated events the chances seem pretty good it has occurred at some point in the past. But since each event in isolation seems likely to have occurred in the past, that would seem to imply that everything has occurred. But that seems less likely, and complicates matters severely, in ways I haven't entirely thought through, and I'd be curious to hear what people think.

For example, the "human" factor hugely complicates things, because for "everything" to have occurred, many people would have had to do irrational things. This, more than anything, makes me think that "everything" has probably not yet occurred.

Nevertheless, even if "everything" hasn't occurred, there should be a way to estimate how long, on average, it should take to occur (using something like "table hours" as a base unit, with one table hour representing a predetermined number of hands).

And yes, this is useless to our poker skill.

But sometimes useless is fun [img]/images/graemlins/grin.gif[/img]

[FlFishOn]

There's maybe 10^30 deals. No doubt many have yet to dealt out.

[VS.Goliath.]

Weird. I was thinking almost the exact same thing today (no joke). What are the chances of that??

I wondered if you could study every possible situation (or groups of similar situations) enough to know exactly what to do every time.

But I would have no idea how to calculate how long it will take for "everything" to happen.

[SacredChao]

Believe it or not I thought about this one time and tried to figure out just the same thing. The number of games is surprisingly small compared to the number of shuffles of the deck. It started out as a thinking exercise about deck shuffling. The way that texas holdem is played the mumber of unique games is much smaller than the number of shuffles.

First let's look at the number of shuffles. This is 52! which is a very large number. So large that if everyone that ever lived dealt a hand every secondfrom a different deck it would take longer than the age of the universe to deal all possible shuffles. I made some estimates on the total population of the earth and even if I was off by a factor of 1000 it doesn't change the length of time to less than the age of the universe.

Now, one thing to remember that when dealing a hand of holdem is that there are cards left over. This means that there are a number of shuffles that are degenerate. By degenerate I mean that if you took the bottom cards and shuffled them creating a unique shuffle, they would all produce the same game. So, for each 10 handed game of holdem there are 27 cards left over. This means that there are 27! shuffles that produce the exact same game. This is also a huge number. It is also a small number compared to 52!. But a lot of small numbers can eventually have a noticeable effect on a large number. It got me thinking about calculating the total number of game situations at a table. I did some calculations and found the number to be surprisingly small.

I also found some other things about the number of games that are dependent on the number of players.

It's a bit late for me right now but I will make another post with some of the results I came up with at a later date. Also I'd like to check my numbers since I did this for my own entertainment and will now be effectively submitting it for critical review.

[nineinchal]

No everything has not happened. When the WSOP incorporates one card shmuck and baseball into the menu then everything would have happened.

[FlFishOn]

[ QUOTE ]
There's maybe 10^30 deals. No doubt many have yet to dealt out.

[/ QUOTE ]

Oops, 10^40 is closer to the truth. Only missed it by ten. More like a ten trillion fold error. WOn't be my biggest today.

[Siegmund]

I make the number of distinct deals (suits matter) as 1.2x10^36 ~ 52!/(27! 2^10 * 6). Each of these, however, can have a very large number of betting sequences laid on top of it.

The expected number of trials to get every member of a set at least once is about N ln N -- so, we need time to deal about 10^38 hands before we have a good chance of having hit every one of them (even if the betting went differently!)

[ QUOTE ]

But since each event in isolation seems likely to have occurred in the past,


[/ QUOTE ]

A simple failure of intuition. The biggest internet sites have dealt only a few billion hands in their lives; the internet sites and live games combined will be a few tens of billions of hands that have ever been dealt. This means that the chance of any given full deal having ever been dealt before anywhere is less than 10^-25.
That in turn means that the expected time until we see our FIRST duplicate hand is more than 10^12... in other words, most of us will live our WHOLE LIVES without EVER playing a single full-table deal that has ever been played before by anyone else.

[SamIAm]

If you're playing heads-up (I know the OP said 10max) and the suits don't matter, I'm sure there have been some identical hands.

Like, I'm sure the scenario "SB gets 2[img]/images/graemlins/club.gif[/img]3[img]/images/graemlins/spade.gif[/img]. BB gets A[img]/images/graemlins/diamond.gif[/img]A[img]/images/graemlins/heart.gif[/img]. SB folds preflop" has happened many times, especially if you can permute the suits.
-Sam

[Siegmund]

Yes, with only a million ways for two people to be dealt hands, the same preflop action has happened many times. With about 55 trillion complete heads-up deals (52!/43!2!2!3!) there are still a great many heads-up hands that have never been seen, but a regular heads-up player will run into ones that have been dealt before somewhere every month or two, and is likely to see a few hands twice over his own lifetime.

[JustBeginning]

Just as an example, has anyone actually hear of this:
K [img]/images/graemlins/spade.gif[/img] Q [img]/images/graemlins/spade.gif[/img] against A[img]/images/graemlins/club.gif[/img] A[img]/images/graemlins/heart.gif[/img] preflop.
Flop: A[img]/images/graemlins/spade.gif[/img] 10 [img]/images/graemlins/spade.gif[/img] A [img]/images/graemlins/diamond.gif[/img] (I think people may see where this is going)
Turn: J[img]/images/graemlins/spade.gif[/img]
River: who cares (but actually this would have to have happened with every single card on the river, so 44 times, not to mention the different stages of the previous cards coming out and different starting hands.)