I guess it depends where it is being held?

Is your answer to your own question going to be ?/52?

Roughly. It is actually closer to 0.58/30.

It is actually a little less than 0.58/30. 0.57692307/30 where the 692307 is a repeating decimal. To keep up the nittery.

I think people have an easier time visualizing the probability when the numerator is a 1 though. So I think generally it would be written with the numerator being a 1 and the denominator being adjusted proportionally.

Hi pk,

So, 1/52 = 0.01923/1 and 0.57692307/30 = 0.01923/1?

Where's he going with this?

My daughter got home from middle school about an hour ago. She asked me why I was so excited. I told her I just discovered a new way of thinking about probabilities that revealed that the traditional method of thinking about and calculating probabilities was wrong!

She gave me that look that all dads dread, that disapproving, embarrassed "I cannot believe that you are my father" look.

Since she has learned basic probability in middle school, she somewhat reluctantly asked me what my new discovery was. I tried to get her to understand but she kept interrupting me with rude questions and what I considered to be snide remarks. She finally threw up her hands and loudly demanded that I show her a calculation that is done wrong the old way and what the newly discovered correct way is.

I wanted to tell her not to yell at her father like that, but I bit my tongue since I didn't want to lose any more time in teaching my daughter the new probability math.

I told her to imagine 30 fully shuffled 52-card decks of regular playing cards sitting face down on our dining room table. I told her to imagine taking the top card, without looking, from each deck so she would now have 30 playing cards. I then asked her to imagine vigorously shuffling those 30 cards. I then told her to imagine taking the top card of the 30 fully-shuffled cards without looking at it. I then asked her what is the probability that that card would be the Ace of Spades.

After thinking about it for about one second, she was about to give her answer when I stopped her. I walked her over to my desk and sketched out the problem on a piece of yellow legal paper. I showed her the x dimension and the y dimension. I pointed out that the x dimension (one deck) would have exactly one Ace of Spades whereas the y dimension (the first card of the 30 decks laminated) could have anywhere from 0 to 30 Aces of Spades. So clearly, the x dimension is fundamentally different from the y dimension.

She was about to say something about that viewpoint when I stopped her again. I then reminded her that she (in our experiment) is now holding 30 cards. And so the probability that the top card of the new deck of 30-top cards from the 30 52-card decks being the Ace of Spades must be some fraction of 30. I triumphantly wrote the following expression at the top of the legal pad and put a big box around it:

P = ?/30

The desired probability we seek, P, I proudly exclaimed, is ?/30.

I said that I really wanted to know what ? was. I grabbed my car keys and asked my daughter if she wanted to go with me. She asked where I was going. To the store, obviously, to buy 30 decks of playing cards. I want to do the experiment and determine ? once and for all.

My daughter's expression turned from bemusement to horror rather than the joyous rapture I was expecting at the prospect of spending all evening with her father shuffling cards and tracking how many Aces of Spades were at the top of the 30-card deck of top cards.

She gently took away my car keys and led me back to the couch in the living room. Sitting me down she said that she could program a "simulation" of the experiment on her laptop in just a few minutes and the program could "simulate" millions of trials of the experiment. Doing it by hand with real cards would be a major "pain" and the "sample size" would be a joke. The way kids talk nowadays!

I said that sounded all well and good, but I asked here isn't there a way to determine ? without doing any "simulations".

She said that there is something called the "bye-no-me-ill" distribution which gives the likelihoods of the various possibilities of the number of Aces of Spades in that 30-card deck. Then she could use those likelihoods to "wait" the probabilities of selecting the Ace of Spades for each case ranging from 0 to 30 Aces of Spades in the 30-card deck. If there are N Aces of Spades in the deck, she continued, the probability of the top card in a fully shuffled deck of 30 cards being the Ace of Spades is N/30. (Kids say the darndest things.)

All this thinking was giving me a headache so I lied down on the couch for a few minutes with my eyes closed. To be honest, I couldn't get those dancing ? out of my head.

I must have dozed off for a time for a little later my daughter was shaking me awake. She had answers to the problem.

Standing and speaking like she was giving a report in front of her middle school class, she cleared her throat and proclaimed the results of the simulation were in.

In a million trials of selecting the top card from our fully shuffled deck of top cards from 30 fully shuffled decks of 52 cards she got 19,250 Aces of Spades.

I asked what does that mean. Does that tell us anything about P, our desired probability?

She rolled her eyes at me and said "Duh, that is the probability". What is, I asked. She said 19,250/1,000,000 or 1.925%. Of course, she added, this is not the exact answer since it was done over only 1 million trials. If the simulation were allowed to run over 10 million or 100 million or even 1 billion trials, the result would be more accurate. Something about some law of big numbers.

Okay, so that is like a really good estimate. Can we do better?

Yes, don't you remember? She said that while the simulation was running, she used the "bye-no-me-ill" formula to derive how often the Ace of Spades would be at the top of the 30-card deck of top cards. She then read me off the results:

55.85% of the time the 30-card deck would have no Aces of Spades, and, of course, the top card of such a deck would never be the Ace of Spades.

32.85% of the time the 30-card deck would have exactly one Ace of Spades, and the top card of such a deck would be the Ace of Spades 1 out of every 30 times.

9.34% of the time the 30-card deck would have exactly two Aces of Spades, and the top card of such a deck would be the Ace of Spades 2 out of every 30 times.

1.71% of the time the 30-card deck would have exactly three Aces of Spades, and the top card of such a deck would be the Ace of Spades 3 out of every 30 times.

I was getting bored and hungry so didn't want to listen to my daughter recite the ramifications of all 31 cases (0-30 Aces of Spades being possible), so I asked her to cut to the chase. She looked at me like I used an expression from a bygone era, but she knew what I meant.

Taking all the cases into consideration, she stated for all to hear, we would expect the top card of the 30-card deck of top cards to be the Ace of Spades 0.5769230769 out of every 30 times. When she wrote the number down for me, she put a line over the "230769" that had absolutely no meaning to me but I was too excited to ask about that silly line.

I gushed my stream of consciousness. What is that number? What does it mean? It doesn't even look like a real number. It sure isn't 1/52 (the number that I secretly feared my traditionally trained daughter might return with).

She patiently explained that the weird 0.5769230769 with the line above it was the exact value for ?.

Before I could express my doubts or ask any other questions, she bulldozed through to her coup de gras.

"And that means that P, the probability that the top card of the deck of 30 top-cards would be the Ace of Spades is 0.5769230769 divided by 30. Which is, drumroll please, 1.923076923% (with a line over the "076923"). Which, it should be noted, agrees with the result from the simulation."

What is that? That doesn't look like any number I've ever seen before. And why does that pesky line have to keep showing up?

"That, my dear father, is another name for 1/52. Which is the answer I was going to give you after thinking about the problem for less than one second. If you understand probability at all, it is clear that the probability of the random top card of a deck formed by randomly selecting the top card of 30 decks being the Ace of Spades is just the probability of any card being the Ace of Spades in a well-shuffled deck. 1 out of 52."

I was too stunned to speak. I must have looked crestfallen since my daughter proceeded to make my favorite dinner and told me that she still loved me even if I sometimes have crazy ideas and embarrass her in front of her friends.

I walked over to my desk and crumpled up the piece of legal paper with my x-y array on it and threw it in the waste basket.

Let's eat, I happily proclaimed, secure in the knowledge that I had a very smart and loving daughter.

A+++++

Yeah, but, (not trying to be nitty like certain others), please ask her how many Aces of Diamonds would there be.

cool. anyways, ps is rigged i flop quads daily.

RESPONSE IS AT THE 1:12 MARK:






But:

is an important question.

Appearing Nightly

The Trolling Mods

'cept when they are not

Ostensibly.

An array only applies if multiple decks are used in a single hand. Which of course they are not. Each hand is 1 deck and 1 deck only.

The flaw here is that you had her imagine what would happen if you used 30 decks of regular playing cards. Of course it is 1/52 with regular playing cards. You have to have her work this out again, only this time imagining 30 decks of randomly arranged virtual playing cards. The result will undoubtedly be ?/30.

You forget, we have an array
↑
and an array
→

Not to be confused with
↕ or ↔

Probably closely related to
♀ and ♂

Mostly though, it probably looks like ?/¿ in pkdk's world

Why are my posts being deleted?

Sorry guys I have answered you all, and somebody has removed them.

*

Edit/MH: It was accidental. Please feel free to answer them again. Preferably in one post instead of in two or more consecutive ones.

When in a game of poker live is there ever only a 50% chance that there is an ace of diamonds in the 52 variables?

When in a game of live poker are you using 30 decks at the same time?

Live poker no, internet 1000,s of decks are dependent to a players game.

Hello Stephen,

Thank you for your email.

While we obviously don't use real (physical) decks, technically, every hand a new deck is used. We have a dedicated shuffle server that does nothing except shuffle millions of decks every day.

Once a new deck has been shuffled the shuffle server does not know which table any of the shuffled decks are going. It just sits there shuffling tens of thousands of decks every minute. These decks are then put in a queue to send to the next table that needs one. The server does not know how many players will be at the table that requests the next deck, it does not know what poker game the cards will be used for, and it does not know the identities of the players. It just shuffles cards.

PokerStars' shuffle software is truly random, and produces a deck which is completely fair and unpredictable.

To do so, it takes two completely and truly random sources of information:

1) Player input (e.g. mouse movements, key strokes, clicks, etc.)

2) A physical random number generator exploiting an elementary quantum optics process

These are combined to create a completely random number. This is then used to shuffle the cards - you can read about exactly how this is done here:

http://www.pokerstars.com/poker/room/features/security/

Details on the Quantis device which we use can be seen at:

http://www.idquantique.com/random-nu...-overview.html

Because nobody can predict either of the two sources of information, the number generated by our Random Number Generator (RNG), and consequently the order of the cards, is completely random in a mathematical sense. The data taken from Player input is not used to shuffle the next hand at the table its is actually used by the shuffle server for shuffling a new deck of cards with an unknown destination.

Confirmed today by PS


Event order

shuffle server → array → new deck to any random table that needs one.

But crucially, there's never more than one deck per hand!

There is crucially only one deck at a table at a time, but on the choice of new deck from the queue , there is millions of decks dependent to the choice of deck by random timing .

So if you can imagine you are looking directly ahead, you can see 52 individual cards, each card having a 1/52 chance of being the top card, nearest value to you .

x→{1,2,3,,,,,,,,,,,,,,,52}


Now if I asked you turn left , you would be now looking at millions of random cards with no idea of what they are .


millions
.
.
.
.
.
↑
y


That is the setup of PS

y
.
.
.
..............x

x ≠ y

→1/52
→1/52
→1/52
→1/52
....↑
..?/?


all x= 1/52

all y = var 1/52


Vector analysis reveals a second set of probabilities y . In example let us imagine that y has 100,000 aces aligned to the SB, the top card of any y . That is 100, 000 chances of an ace on the SB.

Hi pk,

Nice. The random generator works randomly and generates completely random decks. So far so good. Probably it works that way for the initial cards dealt. However, once the players hold their cards, the board cards can be served in a "slightly less than random" way. For example, the next card can be randomly chosen not out of the remaining x cards, but out of x-n cards, so that the board benefits certain account which is in the hand, say the account S. Like you (the mark) have 77, account S has Ah9h and the board reads As9s7h2d, the river is then drawn randomly from a smaller set of cards, like Ac,Ad,9c,9d. How do we know that the software does not use profiling of players based on things like how far is each non-site account from his or her last deposit? (By a non-site account I mean non-prop, non-bot, non-site employee.)

if you can demonstrate 4 possibilities, that is not the two possibilities of 1/2

Because we get a random deck, it is impossible to target any specific player.