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Chicago Poker» Forums » Rules

Subject: Error in the hand ordering? rss

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Mark Rigby-Jones
United Kingdom
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As I occasionally enjoy playing with numbers, and poker hand probabilities in particular (especially after catching an error in a book on probabilities), I thought I'd take a look at the odds in Chicago Poker. WARNING: Maths ahead!

The easiest way to calculate the probabilities is to enumerate the possible hands of a type, and divide by the total number of hands - here, this is 75 choose 5 (I'll use the notation 75c5), or 17,259,390.


CHICAGO POKER: 15 possible numbers
15 hands, 1/1,151,087

STRAIGHT FLUSH: 11 start cards, 5 colours
11 x 5 = 55 hands, 1/313,933

RAINBOW STRAIGHT: 11 start cards, 5 factorial colour combinations
11 x 5! = 1,320 hands, 1/13,081

FOUR OF A KIND: 15 numbers, 5 missing colours, 70 other cards
15 x 5 x 70 = 5,250 hands, 1/3,289

FULL HOUSE: 15 3-card sets, 14 2-card sets, each with colour combinations
15 x 14 x 5c3 x 5c2 = 21,000 hands, 1/822

FLUSH: 5 colours, numerous number combinations, minus straight flushes
5 x 15c5 - 55 = 14,960 hands, 1/1,154

STRAIGHT: 11 start cards, numerous colour combinations, minus rainbows and flushes
11 x 5^5 - 1320 - 55 = 33,000 hands, 1/523

THREE OF A KIND: 15 numbers, various colour combos, two other cards, minus full house
15 x 5c3 x 70c2 - 21000 = 341,250 hands, 1/51

TWO PAIR: 15 first number, 14 second number, colour combos, 65 other cards
15 x 14 x 5c2 x 5c2 x 65 = 1,365,000 hands, 1/13

ONE PAIR: 15 pairs, various colours, three of 70 other cards, minus two pair and full house
15 x 5c2 x 70c3 - 1365000 - 21000 = 6,825,000 hands, 1/2.5

HIGH CARD: Whatever's left
Total - all others = 8,652,540 hands, 1/2


Assuming I've done my sums correctly, that means that the order of full house and flush should be reversed - the different number of cards and suits throws off the odds compared to the standard poker order.

Compared to standard poker, the probabilities of high card, pair and 3 of a kind are about the same. Two pair and four of a kind are both more likely with Chicago Poker, all other hands are less likely.
 
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Dana More
United States
Garden Grove
California
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Thanks for bringing this up. My math agrees with yours, FWIW. I can't believe nobody bothered to check the probabilities before releasing the game. I guess they figured nothing would change enough to alter the ranks of the hands, despite the differing card composition.

I'll definitely be changing this when I play, so that a flush beats a full house...
 
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Greg Case
Canada
Waterloo
Ontario
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zorac wrote:

Compared to standard poker, the probabilities of high card, pair and 3 of a kind are about the same. Two pair and four of a kind are both more likely with Chicago Poker, all other hands are less likely.


In standard poker the Ace also has a dual role as 1 and 14
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