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Subject: A Geek of Math's complaint (or strategic hint) rss

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Matthieu Weber
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OK, I'm not really a geek of maths, but I do have a heavy past related to math. Also, I've never been good at combinatory analysis, but when I saw the different card combinations that you can use in the game, I simply had to calculate the odds. If you don't grok maths and decide to just trust me (you shouldn't), jump to the next paragraph.

So here we go:

1. Straight Flush: 7 x 6 = 42 combinations
2. Three of a kind: 54 x 5 x 4 = 1080 combinations
3. Flush: 54 x 8 x 7 = 3024 combinations
4. Straight: 42 x (6 x 6 - 1) = 1470 combinations

What does it tell us? Obviously, there are many more ways to build a flush than a straight. Therefore, the straight should win over a flush. The otherway round, it means that when playing the game, you should forget completely about Straights and concentrate in building Flushes and higher combinations.

It surprises me that Knizia, who has a Ph.D. in Maths, left such a flaw in his game. Unless I miscalculated the combinations, of course (see the disclaimer in the first paragraph). Any opinions (about my math, Knizia's reasons for keeping the order of the combinations that way, or whatever else)?
 
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Chris
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For the flush:
1. Choose a suit 6 different ways.

2. Chose cards in that suit in no particular order:
9 choose 3 = 9! / (6!*3!) = 84

3. Total number of ways: 84*6 = 504

---
I think your numbers above assume order matters, i.e., a permutation.
 
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Larry Levy
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The Three of a Kind calculation is incorrect as well:

9 ranks x (6 choose 3) [no. of ways to choose from 6 suits] =
9 x 6!/(3!x3!) = 9 x 20 = 180

Chris also made a mistake in his flush calculation (he forgot to subtract off the straight flushes.) So the correct number is 504 - 42 = 462.

So that gives us the following combinations:

Straight Flush - 42
Three of a Kind - 180
Flush - 462
Straight - 1470

As usual, the Good Doctor got it right!
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Chris
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Larry Levy wrote:
Chris also made a mistake in his flush calculation (he forgot to subtract off the straight flushes.) So the correct number is 504 - 42 = 462.

D'oh! I knew something looked off but I couldn't figure out what. Thanks for setting my mind at ease.
 
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Stuart Perry
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Just a thought though....... why would it automatically follow that the harder the pairing (flush, straight, etc...) the higher it ranks in the game? I know that's how all the games I can think of work, but it could (and I stress "could" since the corrected math above shows it is not the case for ST) be an intentional twist. Are there any games that work like that?

I know it seems counter-intuitive......
 
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Tony Chen
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Eugene van der Pijll
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noursy wrote:
Obviously, there are many more ways to build a flush than a straight.



Yes there are, but the number of possible combinations is not the only factor here. When you're starting a new combination on your side, you will mostly try to go for a straight flush or a three of a kind. Flushes and straights are only options if those combinations are made unavailable. Most often, this will happen after two cards of a straight flush have been played, and the other card (or cards) is played by the opponent.

For example, if you have played the green 2 and 4, and the opponent has played the green 3, you may play one of the other 5 3s to make a straight, or one of the other 6 green cards to make a flush. As you see, the odds are nearly even in this case. If you have played the red 5 and 6, and the opponent has played the 4 and 7, you can make a straight with one of the 10 other 4s and 7s, or a flush with one of the other 6 red cards. So in this case, it's easier to make a straight.

So, given that you only go for flushes and straights when the higher combinations have failed, and given that it is on average easier to turn a failed straight flush into a straight than into a flush, flushes should be ranked higher than straights.
 
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Tony Chen
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"yes there are"
no there aren't

 
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Matthieu Weber
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Damn.

Well, I did warn that I sucked at that kind of math

Thanks for the corrections!
 
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sunday silence
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terp8in wrote:
Just a thought though....... why would it automatically follow that the harder the pairing (flush, straight, etc...) the higher it ranks in the game? I know that's how all the games I can think of work, but it could (and I stress "could" since the corrected math above shows it is not the case for ST) be an intentional twist. Are there any games that work like that?

I know it seems counter-intuitive......


I believe poker itself would meet your question. Originally four of a kind ranked higher than a straight flush (I think it is harder to attain). HOwever when holding four aces, you are certain of victory and this was thought somehow to be ungentlemanly, at least that is how Parlett explains it.

So they made a straight flush higher than four of a kind out of some sort of fairness issue. ANd so it is possible that with a straight flush you might still only split the pot (with another straight flush, as remote as that seems)
 
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Russ Williams
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sundaysilence wrote:
I believe poker itself would meet your question. Originally four of a kind ranked higher than a straight flush (I think it is harder to attain). HOwever when holding four aces, you are certain of victory and this was thought somehow to be ungentlemanly, at least that is how Parlett explains it.

So they made a straight flush higher than four of a kind out of some sort of fairness issue. ANd so it is possible that with a straight flush you might still only split the pot (with another straight flush, as remote as that seems)

Hmm, that seems inconsistent with http://en.wikipedia.org/wiki/Poker_probability

Probability of 4 of a kind = 0.0240%
Probability of a non-royal straight flush = 0.00139%
Probability of a royal flush = 0.000154%

So 4 of a kind is not harder to attain than a straight flush.
 
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sunday silence
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Yes I see your right, and so I reach back for my Parlett: A History of Card Games. on pp 113-114. The passage is rather contorted as he says

" The problem with straights, here ranked in wrong order [he's referring to an 1864 Hoyle saying the straight is ranked in the wrong order] was partly mathematical but largely one of principle.." he goes onto quote some letter to the editor who claims straights are more likely than 3 of a kind, which they are not finally he says:

"If you ignore straights, and hence straight flushes too the highest possible hand is a top four of a kind which is not just unbeatable but cannot even be tied."

So the pt. was that some people didnt think the game should have straights (old poker was based on four cards and it didnt have straights) but the new generation found straights more interesting as well as the bit about four aces being unbeatable and not nice. But the passage is confusing as hell as he states in the part above that they are ranked in the wrong order and then on the next page he continues:


"In this light the acceptance of straights ranked in the wrong order may be seen as a temporary compromise..."

So I remain confused. I am not sure what he means when he says they are ranked in the wrong order. or maybe they were in the early days. Hell doesnt Parlett have an email? I think he does.
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sunday silence
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Here is a website from the "gaming guru" a retired professor and he has an article on four card poker: Note the rankings in "Game 2" which shows the rankings based on probability of a four card hand:

http://catlin.casinocitytimes.com/article/four-card-poker-pa...

So apparently this all has to do with four card poker hands, later in the article he is talking about making four card hands out of five cards so I am more confused.

Perhaps more interesting is at the end when he notes: "When four-card Poker hands are selected from fiver- card hands, it is impossible to rank the Straights and Flushes in an order than reflects their natural frequencies. Fascinating! "

this owing to the idea that you will group the cards in the best order so no matter which way you rank them, they will not reflect the frequency. Again this is sort of tangential to the main pt.

This article is near verbatim copy of Parlett's passage, the one I was quoting:

http://www.4gamesters.com/history-poker.htm

It's interesting I blame Parlett for not really making any of this clear.


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