Avg EV for 2nd hand dealt from shoe - higher than the first hand?
June 4th, 2015 at 2:48:06 PM
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Given: One player at the BJ table, versus the dealer starting with a fresh shoe. Whatever BJ rules you want although we can define something standard such as 6-decks, H17, DAS, Resplit 4 times - whatever.
Question:What is the EV of the 2nd hand played by the player? Is it higher or lower than the EV of the 1st hand and by how much?
Comments: Before you respond "the EV is the same for all hands, unless we have specific knowledge of the cards that have come out previously" let me warn you that such a response is NOT correct. I am asking a very subtle question.
Look, if we knew the first hand involved exactly four cards to the player and dealer and that those four cards could be chosen with complete randomness, then YES, the 2nd hand EV would be identical to the 1st hand. But that is not the case. The composition of cards that can come out in a BJ hand is constrained by the rules on the dealer and by Basic Strategy rules for the player. These constraints lead to a very complex probability distribution for card composition in the first hand played out of a shoe (or any hand for that matter):
- a minimum of 4 cards, (combinations such as TT vs TT or 98 vs AT)
- a theoretical maximum of 22-23 cards (A2AAA6AAAA5 vs similar hand for dealer) for un-split hands and even larger numbers for hands that are split and re-split
- a capacity for more cards of low rank to be 'consumed' than cards of a high rank
- a hand with exactly 4 Tens coming out would be possible (and indeed, would occur with high frequency) but a hand with four 4's (and no other cards) or 4 8's (and no other cards) are 'forbidden" or impossible.
Given the universe of all the possible compositions of first hands from a fresh shoe, it seems to me that the 2nd hand dealt from the shoe must almost certainly have a different average EV than the first, fresh-shoe hands. As a mathematical curiosity, I am just wondering what the EV of the 2nd hand is and how it differs from the 1st. I'm assuming this would need to be addressed by Monte Carlo simulation, which is not a capability of mine.
I am not expecting to base an AP strategy on the answer to this question - but this question has lodged itself in my brain, and I'm curious to know the answer.
Question:What is the EV of the 2nd hand played by the player? Is it higher or lower than the EV of the 1st hand and by how much?
Comments: Before you respond "the EV is the same for all hands, unless we have specific knowledge of the cards that have come out previously" let me warn you that such a response is NOT correct. I am asking a very subtle question.
Look, if we knew the first hand involved exactly four cards to the player and dealer and that those four cards could be chosen with complete randomness, then YES, the 2nd hand EV would be identical to the 1st hand. But that is not the case. The composition of cards that can come out in a BJ hand is constrained by the rules on the dealer and by Basic Strategy rules for the player. These constraints lead to a very complex probability distribution for card composition in the first hand played out of a shoe (or any hand for that matter):
- a minimum of 4 cards, (combinations such as TT vs TT or 98 vs AT)
- a theoretical maximum of 22-23 cards (A2AAA6AAAA5 vs similar hand for dealer) for un-split hands and even larger numbers for hands that are split and re-split
- a capacity for more cards of low rank to be 'consumed' than cards of a high rank
- a hand with exactly 4 Tens coming out would be possible (and indeed, would occur with high frequency) but a hand with four 4's (and no other cards) or 4 8's (and no other cards) are 'forbidden" or impossible.
Given the universe of all the possible compositions of first hands from a fresh shoe, it seems to me that the 2nd hand dealt from the shoe must almost certainly have a different average EV than the first, fresh-shoe hands. As a mathematical curiosity, I am just wondering what the EV of the 2nd hand is and how it differs from the 1st. I'm assuming this would need to be addressed by Monte Carlo simulation, which is not a capability of mine.
I am not expecting to base an AP strategy on the answer to this question - but this question has lodged itself in my brain, and I'm curious to know the answer.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
June 4th, 2015 at 3:46:35 PM
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Quote: gordonm888Look, if we knew the first hand involved exactly four cards to the player and dealer and that those four cards could be chosen with complete randomness, then YES, the 2nd hand EV would be identical to the 1st hand. But that is not the case. The composition of cards that can come out in a BJ hand is constrained by the rules on the dealer and by Basic Strategy rules for the player. These constraints lead to a very complex probability distribution for card composition in the first hand played out of a shoe
Jadda jadda.... Of course the probability distribution, given the first played hand, is complex. That doesn't mean "oh this is complex, so the answer to my question must be complex too ...".
It is quite simple actually. As long as the second player plays with the same strategy irrespectible what the first player got as cards, his EV will be exactly the same.
If the first player would play face down, what would be the EV of the second player ? Of course it would be the same, unless he uses some complex strategy involving the play of the first player as additional information.
Other than that (if you do respect the additional information): For a single deck game, knowledge of a single random card not involved in your hand gives you about 0.02% advantage for perfect strategy.
June 4th, 2015 at 4:07:47 PM
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Your bold font is obnoxious. It's easy to just delete those "bold" tags, you should do it.Quote: gordonm888I am not expecting to base an AP strategy on the answer to this question - but this question has lodged itself in my brain, and I'm curious to know the answer.
Exploiting changes in strategy (and hence edge) for hands towards third base is known as depth-charging.
The problem you asked, in particular, was solved by a mathematician who worked for Bally's in the early 2000s during the design phase of their "Multi-Hand Blackjack" slot. She is a good friend and is very proud of her work, which was GLI approved (which honestly, doesn't mean that much). Even the two-handed solution is a huge problem if you want an exact answer, and I have no idea how she solved it. And, obviously, I don't know the answer to your question.
The easy solution is to simulate one hand, then tally up the EORs from that hand to estimate the edge on the second hand. Shuffle, rinse and repeat. You don't need to compute anything about the second hand to get a rough idea. You could probably extend this method to a third hand and do it rough justice, as long as the first two hands were played according to basic strategy. I could do this pretty quickly, but honestly, I'd rather watch my dog sleep.
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June 4th, 2015 at 6:15:55 PM
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Quote: teliotYour bold font is obnoxious. It's easy to just delete those "bold" tags, you should do it.
Done, sorry about that.
Quote: teliotExploiting changes in strategy (and hence edge) for hands towards third base is known as depth-charging.
Okay, but I honestly expect the changes in EV will be in the 4th decimal place (or smaller) and I had zero intention of exploiting this. I am just a bit of a theoretical geek and I am honestly just curious. I'd never heard anyone mention that the 2nd hand from a shoe will have a microscopically different EV than the 1st hand and when the underlying principles occurred to me I thought it was interesting and so I thought I would bring it to the attention of the forum and ask about it.
Quote: teliotThe problem you asked, in particular, was solved by a mathematician who worked for Bally's in the early 2000s during the design phase of their "Multi-Hand Blackjack" slot. She is a good friend and is very proud of her work, which was GLI approved (which honestly, doesn't mean that much). Even the two-handed solution is a huge problem if you want an exact answer, and I have no idea how she solved it. And, obviously, I don't know the answer to your question.
The easy solution is to simulate one hand, then tally up the EORs from that hand to estimate the edge on the second hand. Shuffle, rinse and repeat. You don't need to compute anything about the second hand to get a rough idea. You could probably extend this method to a third hand and do it rough justice, as long as the first two hands were played according to basic strategy. I could do this pretty quickly, but honestly, I'd rather watch my dog sleep.
Okay, well at least that's helpful. When I wrote this note I thought it would be over the heads of many of the people here, but I was counting on either you, teliot, or the Wizard understanding what my point was - so I appreciate your reading this and verifying it is some kind of real effect. Clearly, the change in EV for a sequence of multiple hands would be much bigger in a single deck game than for a six-deck shoe.
The simulation approach you mention is the only approach that I could envision- simulate the player's hand and the dealer's hand and come up with a probabilistic table of "true count" and then convert that into a probabilistically weighted EV for the remaining composition of the deck. Or, perform a simulation of millions of cases of multiple sequential hands from a shoe and keep track of the win/loss/push rates as a function of whether the hand was the 1st, 2nd, 3rd or nth hand to be played. A spreadsheet approach seems pretty impossible.
Compliments on having such an interesting dog!
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
June 4th, 2015 at 6:28:16 PM
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Quote: gordonm888Compliments on having such an interesting dog!
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June 4th, 2015 at 6:29:56 PM
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Quote: MangoJ
It is quite simple actually. As long as the second player plays with the same strategy irrespectible what the first player got as cards, his EV will be exactly the same.
** sigh ** No, Mango, I'm sorry but you are wrong. See teliot's response, above, for some corroboration.
The second hand played from a shoe is not played from a fresh deck. It is a fresh deck with one extra piece of information available to you - a single hand has already been played. Since the distribution of cards that can be removed from a BJ deck in one hand is NOT random - more Aces or Twos will be removed on average than the number of Nines, for example - the remaining composition of the shoe is altered, which alters the EV of any hands that follow. We can identify that the EV of the 2nd, 3rd and nth hands are probably bigger than the EV of the first hand as averaged over many many trials -even without specific knowledge of the specific cards that get played in the first hand of any shoe.
Open your mind. This is something new I am identifying - although it has previously been identified and analyzed by an industry mathematician according to teliot.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
June 4th, 2015 at 6:39:19 PM
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New to you. Arnold Snyder talked about it in his book "Blackbelt in Blackjack."Quote: gordonm888This is something new I am identifying.
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June 4th, 2015 at 8:07:21 PM
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Quote: teliot
That Gumby looks almost "new". Very little chewing on that toy. Nice pic!
June 4th, 2015 at 10:26:23 PM
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You do have a beautiful dog, teliot.
As far as the edge change for 2nd and/or future hands, I don't know the answer...other than I'm almost certain that the HE decreases you play more hands (ie: depth-charging).
As far as the edge change for 2nd and/or future hands, I don't know the answer...other than I'm almost certain that the HE decreases you play more hands (ie: depth-charging).
June 5th, 2015 at 12:54:36 AM
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Quote: teliot
Murphy would like to come over and play with Rosie and Gumby! She's ready for your slightest move, there.
If the House lost every hand, they wouldn't deal the game.
June 5th, 2015 at 7:03:14 AM
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Quote: MangoJIt is quite simple actually. As long as the second player plays with the same strategy irrespectible what the first player got as cards, his EV will be exactly the same.
I don't think this was the OP's question - I think the OP was thinking more along the lines of, if you're the only one at the table, once you and the dealer play out a hand, is there any change in the EV for your next hand?
My gut answer is, "If there is, it's negligible," although I'm not as confident about that if it's a single-deck game. It looks like something that needs to be simulated. If I can find time from my literary backlog - right now, I'm at the point of the Song Of Ice & Fire Series where
Tyrion marries Sansa
June 5th, 2015 at 7:50:21 AM
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If the game is played with a cut card, then yes, the EV changes towards the house due to the cut-card effect (assuming a basic strategy player who is not correlating his wagers with the count).Quote: ThatDonGuyI don't think this was the OP's question - I think the OP was thinking more along the lines of, if you're the only one at the table, once you and the dealer play out a hand, is there any change in the EV for your next hand?
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June 5th, 2015 at 7:54:52 AM
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You have already placed your bets, so the only thing you can do is adjust your play decisions - composition dependant.
How much do those composition-dependant plays change by the 0 to several (certainly not many) cards that the 2nd player gets to see before making a decision that the first player did not get to see first?
How much do those composition-dependant plays change by the 0 to several (certainly not many) cards that the 2nd player gets to see before making a decision that the first player did not get to see first?
June 5th, 2015 at 7:58:56 AM
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In single deck, there can be quite a significant change. In a six-deck game, not so much. One of the members here told me the story of how he got backed off when he was flat betting on a single deck game (the old -0.18% games) because of how he changed his playing strategy between hands.Quote: Dalex64You have already placed your bets, so the only thing you can do is adjust your play decisions - composition dependent.
How much do those composition-dependent plays change by the 0 to several (certainly not many) cards that the 2nd player gets to see before making a decision that the first player did not get to see first?
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June 5th, 2015 at 12:10:06 PM
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I think some of the posters misunderstood the question.
I think the question is waht is the Ev of the Second Hand (after a full play of 1 player v dealer) without using any of the information on the first hand to adjust play (ie Standard BS Play).
Assuming BS play on first hand,
I agree with gordonm888 that the Ev will be different even though the difference will be miniscule. So this is just a math curiosity issue only.
And it is obvious that it will be different because a different number of cards drawn depending on the cards drawn.
A though experiment:
You have a Gazillion shoes with all the possible sequence of hands and you start playing.
If you just burned 4 cards and started the '2nd hand' from the 5th for all shoes then you have Exactly the same Ev as '1st' hand.
But
Shoe 1: first 4 card are: 10,10,10,10. So 2nd hand starts on card 5.
Shoe 2:first 4 cards are:2,2,2,2. So 2nd hand does not start on card 5. It starts say on card 7.
So playing through all of the shoes, some will start on card 5, card 6 etc.
Hence 2nd hand will have some biases and hence slighly different Ev (miniscule amount)
I have absolutely no idea whether the difference in Ev will be positive or negative, only that it is miniscule.
The answer can come from Simulation, huge simulation as you are trying to find miniscule differernces.
OR Full Combinatorial.
Full combinatorial should not take that long with some shortcuts.
I think the question is waht is the Ev of the Second Hand (after a full play of 1 player v dealer) without using any of the information on the first hand to adjust play (ie Standard BS Play).
Assuming BS play on first hand,
I agree with gordonm888 that the Ev will be different even though the difference will be miniscule. So this is just a math curiosity issue only.
And it is obvious that it will be different because a different number of cards drawn depending on the cards drawn.
A though experiment:
You have a Gazillion shoes with all the possible sequence of hands and you start playing.
If you just burned 4 cards and started the '2nd hand' from the 5th for all shoes then you have Exactly the same Ev as '1st' hand.
But
Shoe 1: first 4 card are: 10,10,10,10. So 2nd hand starts on card 5.
Shoe 2:first 4 cards are:2,2,2,2. So 2nd hand does not start on card 5. It starts say on card 7.
So playing through all of the shoes, some will start on card 5, card 6 etc.
Hence 2nd hand will have some biases and hence slighly different Ev (miniscule amount)
I have absolutely no idea whether the difference in Ev will be positive or negative, only that it is miniscule.
The answer can come from Simulation, huge simulation as you are trying to find miniscule differernces.
OR Full Combinatorial.
Full combinatorial should not take that long with some shortcuts.
June 5th, 2015 at 12:56:21 PM
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The answer is that the edge moves slightly towards the house if a cut card is used.Quote: AceTwoI think some of the posters misunderstood the question.
I think the question is waht is the Ev of the Second Hand (after a full play of 1 player v dealer) without using any of the information on the first hand to adjust play (ie Standard BS Play).
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June 5th, 2015 at 1:00:50 PM
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Quote: teliotThe answer is that the edge moves slightly towards the house if a cut card is used.
That's why I prefer to hear "off the top house edge."
DUHHIIIIIIIII HEARD THAT!
June 5th, 2015 at 1:13:30 PM
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It's going to have to be real big numbers. I ran a quick simulation with 10m shoes where it only played two head-to-head coups per 4-deck shoe (UK rules) (I know this wasn't your question, but given what I already had was the easiest to run).Quote: AceTwoThe answer can come from Simulation, huge simulation as you are trying to find miniscule differernces.
10m shoes - 1st hand : -0.4176% (Win: 4515192 Lose: 5238662 Tie: 877460 BJk: 454472)
10m shoes - all hands: -0.4107% (Win: 9029794 Lose: 10474992 Tie: 1756155 BJk: 908702)
10m shoes 66% penetration (249m hands) -0.4409%
10m shoes 83% penetration (311m hands) -0.4252%
June 5th, 2015 at 3:47:36 PM
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Quote: gordonm888** sigh ** No, Mango, I'm sorry but you are wrong. See teliot's response, above, for some corroboration.
Well, if you read carefully I basically said "as long as the 2nd players strategy ignores the cards of the first player". If you don't ignore the exposed them, obviously you can improve your EV. The resulting strategy is much more complex. If you do ignore it, you won't improve EV.
Quote: gordonm888Open your mind. This is something new I am identifying - although it has previously been identified and analyzed by an industry mathematician according to teliot.
** sigh ** I already did that.... i.e. the analysis of depth charging on a single deck game. The result is stated in #2 of the post. About 0.02% EV for the first exposed card, and actually increases for every other exposed card.
Please don't call me closed minded, I did that calculation already 5 years ago.
June 6th, 2015 at 10:12:37 AM
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Quote: charliepatrickIt's going to have to be real big numbers. I ran a quick simulation with 10m shoes where it only played two head-to-head coups per 4-deck shoe (UK rules) (I know this wasn't your question, but given what I already had was the easiest to run).
As you can see the figures aren't conclusive, you cannot see a consistent pattern based on cards used for first coup, except the ones where three cards were used (i.e. BJ op small card) -0.837%. The most popular was a coup using up 5 cards. In one shoe the first coup used up 27 cards, and then the second one lost!10m shoes - 1st hand : -0.4176% (Win: 4515192 Lose: 5238662 Tie: 877460 BJk: 454472)
10m shoes - all hands: -0.4107% (Win: 9029794 Lose: 10474992 Tie: 1756155 BJk: 908702)
10m shoes 66% penetration (249m hands) -0.4409%
10m shoes 83% penetration (311m hands) -0.4252%
Thank you CharlieP, this is very responsive to what I was wondering about. A few quick questions:
1. What is a "coup?" I am assuming that "two head-to-head coups" are two sequential "player vs dealer" hands that are each dealt to completion. Is that correct?
2. I interpret that in your 10m shoes simulation that the house edge was 0.4176% on the 1st hand and that the house edge diminished to 0.4038% on the 2nd hand so as to yield an average house edge across both hands of 0.4107. Is that a correct interpretation?
I think there are two related but somewhat distinct questions that you are addressing with your MC calcs.
- Over millions of trials, how does average house edge change versus fixed points in a shoe?
- Over millions of trials, is the house advantage on the 2nd dealt hand different that the first?
This latter question is somewhat different because it references a "2nd hand" that is often preceded by 4 or 5 cards removed from the deck but sometimes is preceded by the removal of 27 cards from the deck (with a strong bias in these cases towards the removal of cards of low rank.)
The feature of your results that is puzzling (or indicative of MC statistics) is that the house edge starts at -0.417% in a fresh shoe, and then rises to a shoe-average of -0.441% at 66% penetration (average of 24.9 hands played) and then declines to a shoe average of -0.425% at 83% penetration (average of 31.1 hands played.) Intuitively, one would assume a monotonic rise or fall in house edge as one progresses through the shoe (although that is just intuition.) I guess that's why you say the results are inconclusive!
Quote: charliepatrickIn one shoe the first coup used up 27 cards, and then the second one lost!
LMAO, I imagine every card counter that reads this will react by shaking their head and saying "That's #@&^% Blackjack for you."
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
June 6th, 2015 at 10:41:03 AM
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Quote: teliotThe answer is that the edge moves slightly towards the house if a cut card is used.
Given that we are discussing a player or players using only Basic Strategy (no card counting):
I know that shoes in which more low cards come out (player-favorable shoes) tend to have fewer hands dealt before the cut card is reached -because the hands tend to involve/consume more cards because of all the low cards. Whereas, the player will tend to play more hands in a shoe where many tens are dealt -because the average hand will tend to involve a smaller number of cards. Overall, the player tends to play more hands per shoe in shoes that are trending unfavorable. Is this the effect you are referring to?
If one thinks of a shoe as presenting an "instantaneous edge" that varies as a function of depth or penetration of the shoe, then the issue of a cut card is irrelevant, correct? Is your statement intended to mean that the instantaneous house edge, as averaged over a large number of trials, tends to move slightly down as normal gameplay takes the table deeper into the shoe?
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
June 6th, 2015 at 11:01:09 AM
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I just want to acknowledge AceTwo who absolutely understood the question I was asking. This has nothing to do with exposed cards that the player gets to see and that can form the basis for changing decisions (an interpretation that jmango seems to be addressing)
It is about the fact that, as averaged over a large number of trials, a "player vs dealer" blackjack hand will consume a different number of low cards (say, with rank 2-5) than Ten-value cards, thus very slightly changing the average edge for the 2nd hand dealt from the shoe. The change in edge is expected to be very small and so far, no one is sure of the direction of the change (with the possible exception of teliot).
It is about the fact that, as averaged over a large number of trials, a "player vs dealer" blackjack hand will consume a different number of low cards (say, with rank 2-5) than Ten-value cards, thus very slightly changing the average edge for the 2nd hand dealt from the shoe. The change in edge is expected to be very small and so far, no one is sure of the direction of the change (with the possible exception of teliot).
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
June 6th, 2015 at 12:14:19 PM
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Thanks for your kind words.
Yes I used the phrase "coup" rather than "hands" to avoid confusion, since my program only deals one player hand and one dealer hand.
Everything else you say is correct - in using 10m hands there is a variance in results. This difference creates enough noise to prevent trends of a minuscule magnitude being proved. This is why many more simulations would be needed. I tried 100m and even this isn't conclusive (it also shows the range of results you get).
1st hand = -0.4019% All hands = -0.4077%
I think the 3-7 numbers agree with what might seem intuitive, however it's also interesting to note that where the first coup used 27 cards, you usually lost the 2nd hand!
Yes I used the phrase "coup" rather than "hands" to avoid confusion, since my program only deals one player hand and one dealer hand.
Everything else you say is correct - in using 10m hands there is a variance in results. This difference creates enough noise to prevent trends of a minuscule magnitude being proved. This is why many more simulations would be needed. I tried 100m and even this isn't conclusive (it also shows the range of results you get).
1st hand = -0.4019% All hands = -0.4077%
| #Cards | E V | Hands | ||
|---|---|---|---|---|
| 0 | -0.4019% | 100 000 000 | -0.4176% | 10 000 000 |
| 3 | -0.6740% | 2 957 262 | -0.8371% | 295 500 |
| 4 | -0.6309% | 22 566 098 | -0.5105% | 2 256 633 |
| 5 | -0.4588% | 35 145 126 | -0.4050% | 3 514 754 |
| 6 | -0.2332% | 25 087 252 | -0.3446% | 2 507 617 |
| 7 | -0.2359% | 10 028 991 | -0.3147% | 1 003 519 |
| 8 | -0.1628% | 2 883 082 | 0.0578% | 288 286 |
| 9 | -0.3176% | 830 322 | -0.4032% | 83 324 |
| 10 | 0.0427% | 282 113 | -0.2344% | 28 154 |
| 11 | -0.2496% | 112 567 | -0.5615% | 11 399 |
| 12 | -0.2542% | 52 514 | 0.0569% | 5 268 |
| 13 | -0.8302% | 26 742 | 0.1099% | 2 730 |
| 14 | -1.2918% | 13 431 | -2.7653% | 1 338 |
| 15 | 3.0720% | 6 722 | 2.5811% | 678 |
| 16 | 0.2792% | 3 582 | -6.5617% | 381 |
| 17 | -0.0775% | 1 936 | -13.9053% | 169 |
| 18 | -2.4651% | 1 075 | 10.8000% | 125 |
| 19 | 4.8845% | 563 | 5.2239% | 67 |
| 20 | 15.7895% | 266 | 0.0000% | 19 |
| 21 | 4.8701% | 154 | -26.3158% | 19 |
| 22 | -3.7634% | 93 | -25.0000% | 8 |
| 23 | 4.0816% | 49 | -33.3333% | 6 |
| 24 | 1.5152% | 33 | -100.0000% | 1 |
| 25 | -10.0000% | 10 | -100.0000% | 1 |
| 26 | -42.8571% | 7 | 33.3333% | 3 |
| 27 | -100.0000% | 3 | -100.0000% | 1 |
| 28 | 0.0000% | 4 | ||
| 29 | 100.0000% | 1 | ||
| 30 | -100.0000% | 1 | ||
| 32 | 100.0000% | 1 |
June 6th, 2015 at 11:25:28 PM
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Way cool, CharliePatrick. Thanks so much. It will take time to absorb this.
I'd love to see results from the couple of other forumites who posted and said that they might do some analyses. Keep it coming.
I'd love to see results from the couple of other forumites who posted and said that they might do some analyses. Keep it coming.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
