Originally posted by shatteredaces
You all are avoiding even looking at the Paradox or considering it or answering my questions, I have provided the cross correlation link, it is not my fault none of you are able to understand it.
The Paradox is 100% there, I would show you but you avoid the simple questions and return with you are wrong when you are not even considering the Paradox., quoting back the present information that we all already know...
shatteredaces:
On the contrary, we have been extremely patient, we have listened to your arguments, and tried to answer your questions.
In your scenario where a single card is drawn from 52 separate standard decks we can mathematically show that the probability of having exactly 4 aces is not 4/52. Is this the paradox?
How is it a paradox? It is simply a mathematical fact.
Originally posted by shatteredaces
answer the very simple questions and I will show you the Paradox.
N
N
N
Three random cards, what is the chance that one of these is an ace?
Options
no aces
1 ace
2 ace
3 aces
do you agree with this?
At the moment you are all saying that the quantity of 3 is 52, and 3 cards are 4 known aces.
Ahem. No one in this discussion has made such a statement. First of all, the phrase "the quantity of 3 is 52" has no meaning in English -- or in mathematics, as far as I know.
You ask:
Three random cards, what is the chance that one of these is an ace?
If the selection is made from a single deck, then the probability that at least one is an ace:
1- (48/52)*(47/51)*(46/50) or 0.217375566 (I used Excel)
If you draw 1 card from each of 3 decks, then the probability that at least one is an ace is:
1- (48/52)*(48/52)*(48/52) which equals 0.213472918.
This is also the probability of drawing at least one ace if you put the drawn card back into the single deck before drawing the next card.
Interestingly, if you mix the 3 decks together before drawing, so that you have a single deck of 156 cards, then the probability of drawing at least 1 ace is:
1- (144/156)*(143/155)*(142/153) = 0.214746544
Originally posted by shatteredaces
4/52 is because we know there is 4 aces in 52 cards.
This is a serious insult to anybodies inelegance.
Did you mean intelligence?
Originally posted by shatteredaces
Pick 5 random cards from 5 random decks, we have 5 cards in total, we throw the rest away, your new choice is of 5, not 52, the amount of aces is now unknown.
Are you all seriously saying that 5 cards is really 52 cards?
In each scenario, the amount of aces is always unknown. The probability is always known, and can always be calculated.
If I pick 5 random cards from 5 random decks, yes we have 5 cards. The other 255 can be thrown away or not -- it is irrelevant. No-one is saying the 5 cards is 52 cards. I don't get why we think this. Can you quote a post where someone says anything like this?
Probability is a mathematical concept.
It has no influence in the physical world, rather it describes the physical world.
The cross-correlation article you reference does indeed mathematically describe the situation you are referring to: The relationships between independent sets over time.
There are many problems with this:
It has no relation to poker, since -- as the article states -- the sets are independent. In poker we deal only with one deck at a time.
In your analysis of decks of cards, time is irrelevant -- whether the sets are ordered all at once, or whether they are ordered with a delay does not matter.
Let us reduce this to the simplest case possible:
Case 1:
We have a set : (A B)
If we choose 1 member of set, then there is a 0.5 probability it is A
Case 2:
We have two identical sets: (A B) (A B)
If we choose 1 member of each set the possibilities are:
(A)(A)
(A)(B)
(B)(A)
(B)(B)
As you can see, the probability that there exactly 1 A is 0.5 (2/4) but the probability of there being at least one A is 0.75
In the physical world, it is necessary to run millions of trials, even in simple situations before the occurrence of real events becomes close to the mathematical probability.
For example: There are 1326 different two-card combinations. 4 of them are KK.
So, in my database I should expect to find the 4/1326 times representing a probability of:
0.00301659125188536953242835595777
My database contains 254,251 NL Holdem hands. Not a lot.
I "should have" had KK 767 times. However I have KK only 760 times.
On the other side, I've had K3o 18 times more than expected.
These things are curiosities, not mysteries
They are mathematical descriptions of real events, if you do the math correctly.
Your "paradox" appears to be a real effect of changing your sample type:
In one scenario you are looking at a set of 52.
In another scenario you are looking at several independent similar sets.
It is little wonder that the probabilities are different. It would be astonishing if they were the same.
You appear to have done the math correctly in each case, but the results are different because the scenarios are different.
No matter how you describe them -- they're still different.
Live with it.
Peace,
VS
