Why are mixed sets worth the most?
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Okay, the math I was using to get the 14/27 value :
After 3 cards : 6/27 = 2/9 probability of a mixed set. There are nine possible hands which make a set, so 18/27=2/3 is the probability of a hand surviving to the 4th card. All hands which survive to the 4th card have the form xxy, with each of the six possible permutations of that form equally likely. Thus, on the 4th card, there is a 1/3 chance to make a mixed set, and a 1/3 chance to make a pure set. Thus, the probability that a mixed set is made on the 4th card is 2/3*1/3 = 2/9. Likewise, 1/3 of all 4-card hands survive to the 5th card, and all surviving 4 card hands have the form xxyy, with each permutation equally likely. Thus, 2/3*1/3 = 2/9 hands survive to the 5th card, and 1/3 of those hands will make a mixed set. Thus, the likelihood of making a mixed set on the 5th card is 2/27. Sum the three ways of making a mixed set and you have the total probability of a mixed set : ie, 2/9+2/9+2/27 = 14/27.
I don't think I'm making any mistakes here, although I'm unsure of why my numbers don't match the output of your program. It is possible that one of my assumptions is off (ie, that the likelihood of each permutation of the surviving four and five card hands is the same).
After 3 cards : 6/27 = 2/9 probability of a mixed set. There are nine possible hands which make a set, so 18/27=2/3 is the probability of a hand surviving to the 4th card. All hands which survive to the 4th card have the form xxy, with each of the six possible permutations of that form equally likely. Thus, on the 4th card, there is a 1/3 chance to make a mixed set, and a 1/3 chance to make a pure set. Thus, the probability that a mixed set is made on the 4th card is 2/3*1/3 = 2/9. Likewise, 1/3 of all 4-card hands survive to the 5th card, and all surviving 4 card hands have the form xxyy, with each permutation equally likely. Thus, 2/3*1/3 = 2/9 hands survive to the 5th card, and 1/3 of those hands will make a mixed set. Thus, the likelihood of making a mixed set on the 5th card is 2/27. Sum the three ways of making a mixed set and you have the total probability of a mixed set : ie, 2/9+2/9+2/27 = 14/27.
I don't think I'm making any mistakes here, although I'm unsure of why my numbers don't match the output of your program. It is possible that one of my assumptions is off (ie, that the likelihood of each permutation of the surviving four and five card hands is the same).
Actually, kev82, now that I think about it, I'm pretty sure that my numbers are more correct that yours.
The problem is, your program gives equal weight to each outcome. This is wrong. Take, as a for instance, that the first two cards are 11. Then, there is a 1/3 probability of getting a 111, a 112, or a 113. However, the way your program is counting, you count exactly one hand which starts 111, 5 hands which start 112, and 5 hands which start 113. Meaning that, according to your program, given that the first two cards are 11, there is only a 1/11 chance of getting that 111 hand.
This is pretty obviously incorrect.
The problem is, your program gives equal weight to each outcome. This is wrong. Take, as a for instance, that the first two cards are 11. Then, there is a 1/3 probability of getting a 111, a 112, or a 113. However, the way your program is counting, you count exactly one hand which starts 111, 5 hands which start 112, and 5 hands which start 113. Meaning that, according to your program, given that the first two cards are 11, there is only a 1/11 chance of getting that 111 hand.
This is pretty obviously incorrect.
But I only count one hand with a 111, because 111 is a set and as soon as we get a set we cash it in and stop.
I produce every card combination up to the point when you cash in, and for every card deal needed, I deal all cards with equal probability (1/3) so I do not see how I have swayed from the initial assumptions.
1) the probability of any colour is 1/3
2) you take cards until you have a set, as soon as you have a set you stop.
I produce every card combination up to the point when you cash in, and for every card deal needed, I deal all cards with equal probability (1/3) so I do not see how I have swayed from the initial assumptions.
1) the probability of any colour is 1/3
2) you take cards until you have a set, as soon as you have a set you stop.
Last edited by kev82 on Sun Mar 11, 2007 3:50 pm, edited 1 time in total.
Well, I don't agree, but I hate combinatorics and probability so I might be wrong. Later I'll write a simulation that does a few million card deals and we'll see what the empirical results are.
I have to apologize, you are right, I still don't see why, but I'll have to have a think about it later.
Here are the results of 100 trials of 10000 card deals
0.5185 0.16047 0.16179 0.15924
0.51937 0.16057 0.15996 0.1601
0.51995 0.15926 0.16171 0.15908
0.51524 0.16148 0.16236 0.16092
0.51589 0.16034 0.16147 0.1623
0.51911 0.16019 0.15958 0.16112
0.5173 0.16067 0.16258 0.15945
0.51685 0.16225 0.16144 0.15946
0.51757 0.16017 0.1615 0.16076
0.51643 0.16093 0.16202 0.16062
0.52095 0.15981 0.16025 0.15899
0.52 0.15939 0.15982 0.16079
0.51977 0.16017 0.1598 0.16026
0.51643 0.16262 0.16033 0.16062
0.51934 0.15926 0.16063 0.16077
0.51842 0.15857 0.16191 0.1611
0.51794 0.16153 0.16022 0.16031
0.51816 0.16038 0.1594 0.16206
0.51757 0.15892 0.16042 0.16309
0.51745 0.16219 0.15988 0.16048
0.52108 0.15942 0.16021 0.15929
0.51956 0.15983 0.15927 0.16134
0.52051 0.16037 0.15825 0.16087
0.51924 0.16069 0.15924 0.16083
0.51886 0.16205 0.1587 0.16039
0.51941 0.16163 0.15925 0.15971
0.51939 0.15975 0.16073 0.16013
0.51697 0.16132 0.16074 0.16097
0.52069 0.1603 0.16133 0.15768
0.51945 0.15963 0.16003 0.16089
0.51856 0.16032 0.15972 0.1614
0.51852 0.15953 0.16058 0.16137
0.52134 0.15952 0.16033 0.15881
0.51712 0.1608 0.16138 0.1607
0.51939 0.15864 0.16037 0.1616
0.5165 0.15994 0.16179 0.16177
0.51875 0.15999 0.1613 0.15996
0.51566 0.16271 0.16124 0.16039
0.51668 0.16189 0.15963 0.1618
0.51997 0.1603 0.15948 0.16025
0.52129 0.16021 0.16098 0.15752
0.5173 0.15968 0.16057 0.16245
0.51852 0.16099 0.1597 0.16079
0.51995 0.15986 0.15866 0.16153
0.51836 0.16074 0.16044 0.16046
0.51848 0.16194 0.15986 0.15972
0.51762 0.16127 0.16062 0.16049
0.51926 0.16075 0.16002 0.15997
0.51997 0.1593 0.15901 0.16172
0.51955 0.1604 0.16037 0.15968
0.51987 0.1607 0.16037 0.15906
0.51452 0.16302 0.16038 0.16208
0.51949 0.16099 0.16029 0.15923
0.51784 0.15853 0.16283 0.1608
0.51747 0.16103 0.15984 0.16166
0.51786 0.15931 0.16186 0.16097
0.52011 0.15999 0.16029 0.15961
0.51973 0.15976 0.15914 0.16137
0.51847 0.16102 0.16002 0.16049
0.51633 0.16387 0.15817 0.16163
0.51896 0.16004 0.15986 0.16114
0.51822 0.15942 0.16117 0.16119
0.51766 0.16063 0.16018 0.16153
0.51895 0.16028 0.16057 0.1602
0.52181 0.16018 0.15955 0.15846
0.51826 0.16114 0.15986 0.16074
0.51715 0.1605 0.16165 0.1607
0.51739 0.16073 0.16084 0.16104
0.51875 0.1585 0.16198 0.16077
0.51938 0.16119 0.1604 0.15903
0.51722 0.1615 0.16014 0.16114
0.5183 0.16195 0.16046 0.15929
0.51761 0.16064 0.15995 0.1618
0.52012 0.15997 0.16088 0.15903
0.51787 0.16194 0.16009 0.1601
0.51917 0.1602 0.16027 0.16036
0.51973 0.16156 0.16083 0.15788
0.51774 0.16141 0.16135 0.1595
0.51922 0.16054 0.15894 0.1613
0.51822 0.16048 0.16001 0.16129
0.51845 0.16265 0.15916 0.15974
0.51915 0.15911 0.16039 0.16135
0.51887 0.15929 0.16104 0.1608
0.52174 0.15816 0.15985 0.16025
0.52015 0.16017 0.16081 0.15887
0.52029 0.15974 0.1623 0.15767
0.51894 0.16124 0.16 0.15982
0.51974 0.16048 0.1586 0.16118
0.51883 0.16229 0.15944 0.15944
0.51676 0.16089 0.16017 0.16218
0.51769 0.16217 0.16117 0.15897
0.51833 0.16019 0.15869 0.16279
0.51769 0.16305 0.15953 0.15973
0.51906 0.15903 0.1619 0.16001
0.5186 0.16047 0.15976 0.16117
0.51909 0.16047 0.15984 0.1606
0.51808 0.16105 0.16149 0.15938
0.52116 0.16053 0.15936 0.15895
0.51944 0.16107 0.15917 0.16032
0.51764 0.1615 0.15859 0.16227
I have to apologize, you are right, I still don't see why, but I'll have to have a think about it later.
Here are the results of 100 trials of 10000 card deals
0.5185 0.16047 0.16179 0.15924
0.51937 0.16057 0.15996 0.1601
0.51995 0.15926 0.16171 0.15908
0.51524 0.16148 0.16236 0.16092
0.51589 0.16034 0.16147 0.1623
0.51911 0.16019 0.15958 0.16112
0.5173 0.16067 0.16258 0.15945
0.51685 0.16225 0.16144 0.15946
0.51757 0.16017 0.1615 0.16076
0.51643 0.16093 0.16202 0.16062
0.52095 0.15981 0.16025 0.15899
0.52 0.15939 0.15982 0.16079
0.51977 0.16017 0.1598 0.16026
0.51643 0.16262 0.16033 0.16062
0.51934 0.15926 0.16063 0.16077
0.51842 0.15857 0.16191 0.1611
0.51794 0.16153 0.16022 0.16031
0.51816 0.16038 0.1594 0.16206
0.51757 0.15892 0.16042 0.16309
0.51745 0.16219 0.15988 0.16048
0.52108 0.15942 0.16021 0.15929
0.51956 0.15983 0.15927 0.16134
0.52051 0.16037 0.15825 0.16087
0.51924 0.16069 0.15924 0.16083
0.51886 0.16205 0.1587 0.16039
0.51941 0.16163 0.15925 0.15971
0.51939 0.15975 0.16073 0.16013
0.51697 0.16132 0.16074 0.16097
0.52069 0.1603 0.16133 0.15768
0.51945 0.15963 0.16003 0.16089
0.51856 0.16032 0.15972 0.1614
0.51852 0.15953 0.16058 0.16137
0.52134 0.15952 0.16033 0.15881
0.51712 0.1608 0.16138 0.1607
0.51939 0.15864 0.16037 0.1616
0.5165 0.15994 0.16179 0.16177
0.51875 0.15999 0.1613 0.15996
0.51566 0.16271 0.16124 0.16039
0.51668 0.16189 0.15963 0.1618
0.51997 0.1603 0.15948 0.16025
0.52129 0.16021 0.16098 0.15752
0.5173 0.15968 0.16057 0.16245
0.51852 0.16099 0.1597 0.16079
0.51995 0.15986 0.15866 0.16153
0.51836 0.16074 0.16044 0.16046
0.51848 0.16194 0.15986 0.15972
0.51762 0.16127 0.16062 0.16049
0.51926 0.16075 0.16002 0.15997
0.51997 0.1593 0.15901 0.16172
0.51955 0.1604 0.16037 0.15968
0.51987 0.1607 0.16037 0.15906
0.51452 0.16302 0.16038 0.16208
0.51949 0.16099 0.16029 0.15923
0.51784 0.15853 0.16283 0.1608
0.51747 0.16103 0.15984 0.16166
0.51786 0.15931 0.16186 0.16097
0.52011 0.15999 0.16029 0.15961
0.51973 0.15976 0.15914 0.16137
0.51847 0.16102 0.16002 0.16049
0.51633 0.16387 0.15817 0.16163
0.51896 0.16004 0.15986 0.16114
0.51822 0.15942 0.16117 0.16119
0.51766 0.16063 0.16018 0.16153
0.51895 0.16028 0.16057 0.1602
0.52181 0.16018 0.15955 0.15846
0.51826 0.16114 0.15986 0.16074
0.51715 0.1605 0.16165 0.1607
0.51739 0.16073 0.16084 0.16104
0.51875 0.1585 0.16198 0.16077
0.51938 0.16119 0.1604 0.15903
0.51722 0.1615 0.16014 0.16114
0.5183 0.16195 0.16046 0.15929
0.51761 0.16064 0.15995 0.1618
0.52012 0.15997 0.16088 0.15903
0.51787 0.16194 0.16009 0.1601
0.51917 0.1602 0.16027 0.16036
0.51973 0.16156 0.16083 0.15788
0.51774 0.16141 0.16135 0.1595
0.51922 0.16054 0.15894 0.1613
0.51822 0.16048 0.16001 0.16129
0.51845 0.16265 0.15916 0.15974
0.51915 0.15911 0.16039 0.16135
0.51887 0.15929 0.16104 0.1608
0.52174 0.15816 0.15985 0.16025
0.52015 0.16017 0.16081 0.15887
0.52029 0.15974 0.1623 0.15767
0.51894 0.16124 0.16 0.15982
0.51974 0.16048 0.1586 0.16118
0.51883 0.16229 0.15944 0.15944
0.51676 0.16089 0.16017 0.16218
0.51769 0.16217 0.16117 0.15897
0.51833 0.16019 0.15869 0.16279
0.51769 0.16305 0.15953 0.15973
0.51906 0.15903 0.1619 0.16001
0.5186 0.16047 0.15976 0.16117
0.51909 0.16047 0.15984 0.1606
0.51808 0.16105 0.16149 0.15938
0.52116 0.16053 0.15936 0.15895
0.51944 0.16107 0.15917 0.16032
0.51764 0.1615 0.15859 0.16227
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tahitiwahini
- Posts: 964
- Joined: Fri Jan 19, 2007 5:26 pm
DuneMe wrote:corner G wrote:in flat rate red is wourth 4, green 6, blue 8, mixed 10. in escallating, it doesn't matter. and the odds are the same for getting three of the same couler, than mixed
May I suggest adding the set values to the FAQ page?
Guess I either need to remember or write it down somewhere.
http://www.conquerclub.com/public.php?p ... tructions3
Or if you use Firefox and stocksr's excellent greasemonkey script the information is available in every game. You can get the script in Utilities and Plug-ins for Conquer Club.
Cheers,
Tahitiwahini
Tahitiwahini
yeti_c wrote:What about considering "not" cashing in your cards...
Sometimes I've had 3 reds and decided that I will attempt to get a mixed by holding fire...
However usually I end up with 3 reds and 2 same colours of even 4 reds...
C.
Believe me, it has been considered and talked about at some length. First problem, the odds of improving are not worth the payout. If you are holding 3 reds, you have a 22% chance of improving to a mixed set by waiting but the pay-off would only be worth a 40% chance. Further, even if you do make that set, you are seeded to be in line for the same crap all over again because you'll have 2 reds left over.
Thus, the only good reason for not cashing in would have to do with some other reason like laying low and hoping everyone leaves you alone in the first round of cash-ins. And that rationale has nothing to do with what kind of set you are holding.
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flashleg8
- Posts: 1026
- Joined: Wed Jan 17, 2007 10:21 am
- Gender: Male
- Location: the Union of Soviet Socialist Scotland
detlef wrote:yeti_c wrote:What about considering "not" cashing in your cards...
Sometimes I've had 3 reds and decided that I will attempt to get a mixed by holding fire...
However usually I end up with 3 reds and 2 same colours of even 4 reds...
C.
Believe me, it has been considered and talked about at some length. First problem, the odds of improving are not worth the payout. If you are holding 3 reds, you have a 22% chance of improving to a mixed set by waiting but the pay-off would only be worth a 40% chance. Further, even if you do make that set, you are seeded to be in line for the same crap all over again because you'll have 2 reds left over.
Thus, the only good reason for not cashing in would have to do with some other reason like laying low and hoping everyone leaves you alone in the first round of cash-ins. And that rationale has nothing to do with what kind of set you are holding.
This is very interesting. I never considered this. I almost always wait before cashing (reds at least) if I don't have a mixed set. Your statistics are compelling though - what are they like for 3 greens? Or blues?
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tahitiwahini
- Posts: 964
- Joined: Fri Jan 19, 2007 5:26 pm
flashleg8 wrote:detlef wrote:yeti_c wrote:What about considering "not" cashing in your cards...
Sometimes I've had 3 reds and decided that I will attempt to get a mixed by holding fire...
However usually I end up with 3 reds and 2 same colours of even 4 reds...
C.
Believe me, it has been considered and talked about at some length. First problem, the odds of improving are not worth the payout. If you are holding 3 reds, you have a 22% chance of improving to a mixed set by waiting but the pay-off would only be worth a 40% chance. Further, even if you do make that set, you are seeded to be in line for the same crap all over again because you'll have 2 reds left over.
Thus, the only good reason for not cashing in would have to do with some other reason like laying low and hoping everyone leaves you alone in the first round of cash-ins. And that rationale has nothing to do with what kind of set you are holding.
This is very interesting. I never considered this. I almost always wait before cashing (reds at least) if I don't have a mixed set. Your statistics are compelling though - what are they like for 3 greens? Or blues?
Aimless does a good job of explaining this here:
http://www.conquerclub.com/forum/viewto ... 326#270326
Cheers,
Tahitiwahini
Tahitiwahini