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Old 04-13-2018, 10:47 AM   #1
buggzilla
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Poker Riddle from FiveThirtyEight.com

This riddle was posted on 538 today. Would love to know what the braintrust here has to say:

You are playing a standard game of Texas Hold ’em, in which you start with two cards. Five community cards will be dealt, and the object is to make the best five-card hand out of the seven cards available to you. However, you’re a perfectionist, and you’re only interested in making the best possible hand once the board has been dealt — such a hand is called the “nuts.” Which two-card starting hand is the most likely to make the nuts once the board has been dealt?

Clearly, preflop, the hand most likely to win hot-and-cold is AA. But how does one approach the problem as stated; i.e., making the best possible hand once the board has been dealt?

Thoughts?
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Old 04-13-2018, 12:23 PM   #2
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Re: Poker Riddle from FiveThirtyEight.com

I've only thought about it for a few seconds, but I think the answer has to be ATs. That way, most of the time you make a flush, it's the nuts; plus you can make the nut straight sometimes; plus the T in your hand blocks straight flushes more often than a higher Broadway card would.

EDIT: Actually now I think it's AKs. It blocks fewer straight flushes, but aces full of kings is the nuts significantly more often than aces full of tens.
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Old 04-13-2018, 02:19 PM   #3
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Re: Poker Riddle from FiveThirtyEight.com

Off the top of my head I'd guess JTs. You can make the maximum amount of straights (which adds up to a ton of boards when you include different suit combinations), all of them are the nut straight (which isn't true for lower connectors), and being suited helps in that you can make straight flushes on flush boards where a regular straight wouldn't be the nuts.
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Old 04-13-2018, 05:46 PM   #4
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Re: Poker Riddle from FiveThirtyEight.com

Quote:
Originally Posted by buggzilla View Post
But how does one approach the problem as stated; i.e., making the best possible hand once the board has been dealt?
Start with figuring out how many of each type of board is possible and determining what those types are.
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Old 04-13-2018, 05:54 PM   #5
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Re: Poker Riddle from FiveThirtyEight.com

Quote:
Originally Posted by buggzilla View Post
This riddle was posted on 538 today. Would love to know what the braintrust here has to say:

You are playing a standard game of Texas Hold ’em, in which you start with two cards. Five community cards will be dealt, and the object is to make the best five-card hand out of the seven cards available to you. However, you’re a perfectionist, and you’re only interested in making the best possible hand once the board has been dealt — such a hand is called the “nuts.” Which two-card starting hand is the most likely to make the nuts once the board has been dealt?

Clearly, preflop, the hand most likely to win hot-and-cold is AA. But how does one approach the problem as stated; i.e., making the best possible hand once the board has been dealt?

Thoughts?

ATs.




This riddle was also posted here in 2010:

https://forumserver.twoplustwo.com/1...?highlight=ats


I said ATs back then too:

Quote:
Originally Posted by Lego05 View Post
My initial thought was AKs or JT (JT I think is clearly wrong though) then I thought some smaller AXs (for some reason A7s popped into my head first) and I'm starting to settle on my guess being ATs.


Someone ran a simulation to get the answer and listed hands in the order that they are most likely to make the nuts. That post is here:

https://forumserver.twoplustwo.com/s...6&postcount=46
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Old 04-13-2018, 07:15 PM   #6
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Re: Poker Riddle from FiveThirtyEight.com

My thought was JTs. The linked thread proves however that CallMeVernon nailed it, ATs.

I had discounted most full houses that are also possible ties with the exact same hand, and forgot those are the nuts as well.
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Old 04-13-2018, 08:34 PM   #7
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Re: Poker Riddle from FiveThirtyEight.com

I thought there was another more recent thread in the Probability Forum on this topic. I remember running many cases on a computer.

Also, I thought that this was NOT a problem in which simulation is useful/needed since a computer can whip through all possible boards fairly quickly to determine how many of them give a given specific two-card hand the absolute nuts (for that board).

Maybe I am remembering some other thread/question.

Edit to add: after skimming the earlier thread, I seem to be very confused what is being asked and how it was answered.

Last edited by whosnext; 04-13-2018 at 08:38 PM. Reason: edit added
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Old 04-14-2018, 02:16 PM   #8
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Re: Poker Riddle from FiveThirtyEight.com

Quote:
Originally Posted by Lego05 View Post
Someone ran a simulation to get the answer and listed hands in the order that they are most likely to make the nuts. That post is here:
https://forumserver.twoplustwo.com/s...6&postcount=46
I'm not too surprised to see ATs, AJs, AQs and AKs are the top four, but I'm intrigued to learn that the next best are apparently A9s and A6s. A6s???
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Old 04-14-2018, 02:24 PM   #9
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Re: Poker Riddle from FiveThirtyEight.com

Quote:
Originally Posted by ArtyMcFly View Post
I'm not too surprised to see ATs, AJs, AQs and AKs are the top four, but I'm intrigued to learn that the next best are apparently A9s and A6s. A6s???


The 9 and the 6 block a ton of straight flushes and give more true absolute nut flushes for the ace, apparently.
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Old 04-15-2018, 05:34 AM   #10
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Re: Poker Riddle from FiveThirtyEight.com

Today I wrote a program to determine the absolute nuts for any 5-card board in NLHE (i.e., the set of hole cards that are guaranteed to win the hand with that board).

I then looped over all possible boards for a few hole cards (one pair of hole cards at a time) and tallied on how many of the total boards possible, given that one player holds those hole cards, do those hole cards make the absolute nuts. Of course, given that a player is dealt a specific pair of hole cards, there are C(50,5) = 2,118,760 possible 5-card boards that need to be considered. So in the table below, 2,118,760 is the denominator for the Pct Board Making Nuts calculations.

Since looping over 2,118,760 possible boards and finding the absolute nuts for each board takes a fair amount of time, I have only so far done the calculations for 11 starting hands. These results are given in the table below.

RankHole CardsBoards Making NutsPct Boards Making Nuts
1
ATs
82,442
3.89%
2
AJs
80,995
3.82%
3
AQs
79,517
3.75%
4
AKs
79,023
3.73%
5
JTs
66,353
3.13%
6
A9s
63,639
3.00%
7
A6s
63,621
3.00%
8
A8s
63,479
3.00%
9
A7s
63,473
3.00%
10
JTo
62,276
2.94%
11
A5s
62,153
2.93%

Time permitting I will try to do another set of starting hands tomorrow and update the table when I have new results.
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Old 04-16-2018, 08:14 AM   #11
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Re: Poker Riddle from FiveThirtyEight.com

Update with more results.

RankHole CardsBoards Making NutsPct Boards Making Nuts
1
ATs
82,442
3.89%
2
AJs
80,995
3.82%
3
AQs
79,517
3.75%
4
AKs
79,023
3.73%
5
JTs
66,353
3.13%
6
A9s
63,639
3.00%
7
A6s
63,621
3.00%
8
A8s
63,479
3.00%
9
A7s
63,473
3.00%
10
JTo
62,276
2.94%
.   
11
A5s
62,153
2.93%
12
A4s
60,694
2.86%
13
A3s
59,204
2.79%
14
A2s
57,653
2.72%
T15
QJs
56,566
2.67%
T15
QTs
56,566
2.67%
17
KTs
55,994
2.64%
18
KJs
55,907
2.64%
19
KQs
55,848
2.64%
T20
QJo
49,676
2.34%
T20
QTo
49,676
2.34%
.   
22
T9s
41,848
1.98%
23
T9o
38,789
1.83%
T24
KQo
37,122
1.75%
T24
KJo
37,122
1.75%
T24
KTo
37,122
1.75%
T27
98s
33,057
1.56%
T27
87s
33,057
1.56%
T27
76s
33,057
1.56%
T27
65s
33,057
1.56%
.   
31
KK
32,750
1.55%
32
AA
31,118
1.47%
T33
98o
30,044
1.42%
T33
87o
30,044
1.42%
T33
76o
30,044
1.42%
T33
65o
30,044
1.42%
37
54s
29,451
1.39%
T38
J9s
28,213
1.33%
T38
T8s
28,213
1.33%
40
AKo
27,044
1.28%
.   
41
54o
26,444
1.25%
T42
J9o
26,189
1.24%
T42
T8o
26,189
1.24%
44
AQo
25,981
1.23%
45
AJo
25,966
1.23%
46
ATo
25,951
1.22%
T47
97s
23,067
1.09%
T47
86s
23,067
1.09%
T47
75s
23,067
1.09%
50
K5s
22,613
1.07%
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Old 04-16-2018, 09:22 AM   #12
ArtyMcFly
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Re: Poker Riddle from FiveThirtyEight.com

Your results are somewhat different to bachfan's. How does KK make the nuts more often than AA?
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Old 04-16-2018, 01:20 PM   #13
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Re: Poker Riddle from FiveThirtyEight.com

Quote:
Originally Posted by ArtyMcFly View Post
Your results are somewhat different to bachfan's. How does KK make the nuts more often than AA?
KK will make with the nuts with top set on a no-straight board more often than AA will (with an A on board, more straights are possible).
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Old 04-16-2018, 07:53 PM   #14
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Re: Poker Riddle from FiveThirtyEight.com

Yes, some of those figures do not agree with bachfan's earlier results. My guess is that I screwed up my logic somewhere.

If I have time I'll try to look into it tonight when, in all likelihood, I'll have to re-run that set of starting hands.
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Old 04-16-2018, 08:07 PM   #15
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Re: Poker Riddle from FiveThirtyEight.com

Quote:
Originally Posted by whosnext View Post
Update with more results.

RankHole CardsBoards Making NutsPct Boards Making Nuts
1
ATs
82,442
3.89%
2
AJs
80,995
3.82%
3
AQs
79,517
3.75%
4
AKs
79,023
3.73%
5
JTs
66,353
3.13%
6
A9s
63,639
3.00%
7
A6s
63,621
3.00%
8
A8s
63,479
3.00%
9
A7s
63,473
3.00%
10
JTo
62,276
2.94%
.   
11
A5s
62,153
2.93%
12
A4s
60,694
2.86%
13
A3s
59,204
2.79%
14
A2s
57,653
2.72%
T15
QJs
56,566
2.67%
T15
QTs
56,566
2.67%
17
KTs
55,994
2.64%
18
KJs
55,907
2.64%
19
KQs
55,848
2.64%
T20
QJo
49,676
2.34%
T20
QTo
49,676
2.34%
.   
22
T9s
41,848
1.98%
23
T9o
38,789
1.83%
T24
KQo
37,122
1.75%
T24
KJo
37,122
1.75%
T24
KTo
37,122
1.75%
T27
98s
33,057
1.56%
T27
87s
33,057
1.56%
T27
76s
33,057
1.56%
T27
65s
33,057
1.56%
.   
31
KK
32,750
1.55%
32
AA
31,118
1.47%
T33
98o
30,044
1.42%
T33
87o
30,044
1.42%
T33
76o
30,044
1.42%
T33
65o
30,044
1.42%
37
54s
29,451
1.39%
T38
J9s
28,213
1.33%
T38
T8s
28,213
1.33%
40
AKo
27,044
1.28%
.   
41
54o
26,444
1.25%
T42
J9o
26,189
1.24%
T42
T8o
26,189
1.24%
44
AQo
25,981
1.23%
45
AJo
25,966
1.23%
46
ATo
25,951
1.22%
T47
97s
23,067
1.09%
T47
86s
23,067
1.09%
T47
75s
23,067
1.09%
50
K5s
22,613
1.07%
Since your percentages are lower than Bachfan's, my 1st thought was you were missing things like both AA and AT can be the nuts on the same board, since AT has a quad blocker, but things appear to be more serious with your app e.g. 76o > 65o, since 76o can make more nut full houses.

Another possible misstep can happen by missing flushes with a str8 flush blocker.

All these possibilities might be causing the lower percentages you are generating. It's tricky.
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Old 04-16-2018, 09:46 PM   #16
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Re: Poker Riddle from FiveThirtyEight.com

Thanks for your comment.

I am definitely capturing (or I am intending to capture) ALL the hands that are the nuts (or effective nuts) on each board. So that is not obviously the reason my code is finding fewer nut boards than Bachfan.

And, yes, I tried to capture Axs being the nuts (the effective nuts) when the x blocks any possible straight flushes. But maybe I screwed that up.

I haven't turned my attention to identifying where my code is screwing up but I do have a question.

It may be obvious, but why "should" 76o make the nuts on more possible boards than 65o can? It is not immediately obvious to me.
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Old 04-17-2018, 01:06 AM   #17
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Re: Poker Riddle from FiveThirtyEight.com

Quote:
Originally Posted by whosnext View Post
It may be obvious, but why "should" 76o make the nuts on more possible boards than 65o can? It is not immediately obvious to me.
There are more boards with 776 and two cards under 6 than there are boards with 665 and two cards under 5.
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Old 04-20-2018, 03:02 PM   #18
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Re: Poker Riddle from FiveThirtyEight.com

FiveThirtyEight gave their answer today. Of course it was JT suited. But they also gave credit - and links - to Two Plus Two Forums for tackling this previously.
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Old 04-20-2018, 03:56 PM   #19
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Poker Riddle from FiveThirtyEight.com

Quote:
Originally Posted by Didace View Post
FiveThirtyEight gave their answer today. Of course it was JT suited. But they also gave credit - and links - to Two Plus Two Forums for tackling this previously.


This must be with the caveat that most full houses aren’t the “solo nuts” ?!

Edit: nvm article answer was ATs.

Last edited by robert_utk; 04-20-2018 at 04:03 PM.
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Old 04-20-2018, 04:10 PM   #20
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Re: Poker Riddle from FiveThirtyEight.com

Yes, typo on my part. Wish I could blame autocorrect but I'm just an idiot.
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Old 04-20-2018, 09:06 PM   #21
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Re: Poker Riddle from FiveThirtyEight.com

I have done only two small things on this topic in the last few days.

1. I have reviewed my code and am appalled and aghast. I over-looked many cases (undercounting) and screwed up the logic in some cases (error in counting). Jeezo Beezo. Not good.

2. I think I now better understand what Bachfan did those many years ago. I was puzzled why he did a simulation over more trials than there were possible boards. If he had the logic to determine the nuts for any board, then he would not need to do this.

Aha. That is the clue. My belief is that Bachfan did not have such logic but he had access to a really fast and powerful NLHE game player that could determine the winner of any NLHE deal (ties included).

So I think that Bachfan generated millions and millions of random boards and by pitting each possible starting hand vs. all other starting hands (he undoubtedly was clever in how he did this) for that board, he could tally how many boards for which each starting hand was the nuts. Maybe this was obvious to everybody except me. To be honest, I have not gone back and reviewed the earlier thread since I have been too busy on other projects. Perhaps this was discussed in the earlier thread.

Anyway, as has been pointed out in this thread above, writing code to determine the nuts for any possible 5-card board in NLHE (using logic, not brute force) is not an easy task. If I have time, I will look into "fixing" my woefully inadequate program over the weekend.
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Old 04-21-2018, 07:14 AM   #22
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Re: Poker Riddle from FiveThirtyEight.com

Don't be too hard on yourself. I wouldn't know where to start with this sort of thing!
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Old 04-21-2018, 11:27 AM   #23
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Re: Poker Riddle from FiveThirtyEight.com

Bachfan is the writer of propokertools, btw
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Old 05-03-2018, 10:18 PM   #24
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Re: Poker Riddle from FiveThirtyEight.com

I finally found the time to do a proper job of tallying on how many possible boards each starting hand in NLHE can make the nuts. My earlier efforts were inadequate and filled with mistakes.

Below is a table of every starting hand and how many NUTS boards it can make with the breakdown of the numbers in each category of nuts (royal flush, straight flush, etc.).

Bachfan previously reported on frequencies based upon simulations of millions of boards for each starting hand. Using the "opposite" approach, I wrote a complex program based upon dozens of logic gates to determine the set of starting hands which would be the nuts for any given 5-card board.

Another program then loops over all possible starting hands, constructs all the possible boards given the starting hand, and then calls the "inner" function that returns all the Nut hands given that board. The "outer" function then tallies for how many boards the specific starting hand is the nuts.

While I have tried to be careful in developing the underlying logic and the copious programs, mistakes are known to creep in (as my earlier attempt surely demonstrated). I hope that everything below is correct but there is always the possibility that something is amiss. Feel free to point out anything that looks suspect.

For comparability (and sanity), the table shows the percentage that I have found via logic-based brute force methods versus the percentage that Bachfan found via his simulations. Statistically, based upon the number of trials he used in his simulations, his results were likely to be within .01 percent of the actual figure.

RankStarting HandRoyal Flush Nut_BoardsStraight Flush Nut_BoardsFour of a Kind Nut_BoardsFull House Nut_BoardsFlush Nut_BoardsStraight Nut_BoardsThree of a Kind Nut_BoardsTOTAL NUT_BOARDSNut_Board PctBachfan_Sim Pct
1
ATs
1,084
135
2,560
4,158
59,850
24,324
0
92,111
4.35%
4.34%
2
AJs
1,084
90
2,575
5,160
56,714
24,324
0
89,947
4.25%
4.24%
3
AQs
1,084
45
2,590
6,246
53,635
24,324
0
87,924
4.15%
4.14%
4
AKs
1,084
0
2,605
7,416
51,013
24,324
0
86,442
4.08%
4.08%
5
A9s
49
180
2,554
3,114
63,607
5,100
0
74,604
3.52%
3.53%
6
A6s
49
133
2,536
882
63,607
5,100
0
72,307
3.41%
3.41%
7
JTs
1,084
3,105
2,071
3,744
0
62,124
0
72,128
3.40%
3.40%
8
A8s
49
133
2,548
2,220
60,897
5,100
0
70,947
3.35%
3.34%
9
A7s
49
133
2,542
1,476
60,897
5,100
0
70,197
3.31%
3.31%
10
A5s
49
1,123
2,530
333
59,850
5,100
0
68,985
3.26%
3.26%
           
11
JTo
94
315
2,116
3,900
230
62,142
0
68,797
3.25%
3.25%
12
A4s
49
1,075
2,533
135
56,714
5,100
0
65,606
3.10%
3.10%
13
KQs
1,084
1,035
2,173
6,120
16,719
36,924
0
64,055
3.02%
3.02%
14
QJs
1,084
2,070
2,101
4,860
3,848
49,524
0
63,487
3.00%
3.00%
15
KJs
1,084
1,080
2,158
5,070
16,808
36,924
0
63,124
2.98%
2.98%
16
QTs
1,084
2,115
2,086
3,930
3,848
49,524
0
62,587
2.95%
2.95%
17
KTs
1,084
1,125
2,143
4,104
16,984
36,924
0
62,364
2.94%
2.95%
18
A3s
49
946
2,536
24
53,635
5,100
0
62,290
2.94%
2.94%
19
A2s
49
0
2,539
0
51,013
5,100
0
58,701
2.77%
2.78%
20
QJo
94
225
2,140
5,040
1,283
49,542
0
58,324
2.75%
2.76%
           
21
QTo
94
270
2,113
4,036
1,045
49,542
0
57,100
2.69%
2.70%
22
KQo
94
135
2,206
6,324
3,408
36,942
0
49,109
2.32%
2.32%
23
T9s
49
4,140
2,050
2,772
0
38,700
0
47,711
2.25%
2.25%
24
KJo
94
180
2,179
5,192
2,568
36,942
0
47,155
2.23%
2.23%
25
KTo
94
225
2,158
4,158
2,327
36,942
0
45,904
2.17%
2.16%
26
T9o
49
313
2,092
2,904
0
38,700
0
44,058
2.08%
2.08%
27
AKo
94
45
2,632
7,644
7,488
24,342
0
42,245
1.99%
1.99%
28
AA
94
0
16,624
0
10,028
5,100
9,300
41,146
1.94%
1.94%
29
AQo
94
90
2,605
6,384
6,164
24,342
0
39,679
1.87%
1.88%
30
AJo
94
135
2,584
5,222
5,317
24,342
0
37,694
1.78%
1.78%
           
31
KK
94
90
16,162
0
4,600
5,100
11,400
37,446
1.77%
1.77%
32
98s
4
3,013
2,038
1,944
0
30,000
0
36,999
1.75%
1.74%
33
ATo
94
180
2,569
4,158
5,070
24,342
0
36,413
1.72%
1.71%
34
87s
4
3,013
2,026
1,260
0
30,000
0
36,303
1.71%
1.71%
35
76s
4
3,013
2,014
720
0
30,000
0
35,751
1.69%
1.69%
36
65s
4
3,013
2,002
333
0
30,000
0
35,352
1.67%
1.67%
37
J9s
49
3,150
2,065
2,934
609
26,100
0
34,907
1.65%
1.65%
38
98o
4
266
2,080
2,052
0
30,000
0
34,402
1.62%
1.63%
39
87o
4
266
2,068
1,344
0
30,000
0
33,682
1.59%
1.59%
40
T8s
49
3,195
2,044
2,082
0
26,100
0
33,470
1.58%
1.58%
           
41
76o
4
266
2,056
780
0
30,000
0
33,106
1.56%
1.56%
42
65o
4
266
2,044
375
0
30,000
0
32,689
1.54%
1.54%
43
J9o
49
268
2,089
3,024
231
26,100
0
31,761
1.50%
1.50%
44
54s
4
3,007
1,999
135
0
26,400
0
31,545
1.49%
1.48%
45
T8o
49
313
2,068
2,156
0
26,100
0
30,686
1.45%
1.45%
46
K9s
49
1,170
2,137
3,222
17,771
5,100
0
29,449
1.39%
1.39%
47
K5s
49
178
2,113
513
20,946
5,100
0
28,899
1.36%
1.36%
48
54o
4
221
2,032
153
0
26,400
0
28,810
1.36%
1.36%
49
K4s
49
133
2,116
207
20,765
5,100
0
28,370
1.34%
1.34%
50
K3s
49
88
2,119
36
20,586
5,100
0
27,978
1.32%
1.32%
           
51
K2s
49
45
2,122
0
20,435
5,100
0
27,751
1.31%
1.31%
52
K8s
49
225
2,131
2,322
17,921
5,100
0
27,748
1.31%
1.31%
53
97s
4
3,058
2,032
1,374
0
21,000
0
27,468
1.30%
1.30%
54
86s
4
3,058
2,020
810
0
21,000
0
26,892
1.27%
1.26%
55
K7s
49
178
2,125
1,566
17,724
5,100
0
26,742
1.26%
1.26%
56
75s
4
3,058
2,008
405
0
21,000
0
26,475
1.25%
1.25%
57
64s
4
3,013
2,005
135
0
21,000
0
26,157
1.23%
1.23%
58
QQ
94
180
15,646
0
2,074
5,100
3,000
26,094
1.23%
1.23%
59
K6s
49
178
2,119
954
17,539
5,100
0
25,939
1.22%
1.23%
60
Q9s
49
2,160
2,080
3,084
4,463
14,100
0
25,936
1.22%
1.23%
           
61
97o
4
266
2,056
1,432
0
21,000
0
24,758
1.17%
1.17%
62
86o
4
266
2,044
852
0
21,000
0
24,166
1.14%
1.15%
63
75o
4
266
2,032
435
0
21,000
0
23,737
1.12%
1.12%
64
64o
4
221
2,029
153
0
21,000
0
23,407
1.10%
1.10%
65
53s
4
3,007
2,002
24
0
17,400
0
22,437
1.06%
1.06%
66
J8s
49
2,205
2,059
2,208
609
14,100
0
21,230
1.00%
1.00%
67
JJ
94
270
15,180
0
460
5,100
0
21,104
1.00%
0.99%
68
Q9o
49
223
2,092
3,130
1,046
14,100
0
20,640
0.97%
0.97%
69
TT
94
360
14,720
0
0
5,100
0
20,274
0.96%
0.96%
70
T7s
49
2,250
2,038
1,476
0
14,100
0
19,913
0.94%
0.94%
           
71
53o
4
176
2,023
28
0
17,400
0
19,631
0.93%
0.93%
72
J8o
49
268
2,071
2,246
231
14,100
0
18,965
0.90%
0.90%
73
T7o
49
313
2,050
1,506
0
14,100
0
18,018
0.85%
0.85%
74
43s
4
1,972
2,005
24
0
13,800
0
17,805
0.84%
0.84%
75
A9o
49
133
2,554
3,114
5,061
5,100
0
16,011
0.76%
0.75%
76
43o
4
131
2,026
28
0
13,800
0
15,989
0.75%
0.75%
77
99
4
266
14,456
0
0
600
0
15,326
0.72%
0.72%
78
Q8s
49
1,214
2,074
2,322
4,468
5,100
0
15,227
0.72%
0.72%
79
A8o
49
133
2,548
2,218
5,066
5,100
0
15,114
0.71%
0.71%
80
88
4
266
14,194
0
0
600
0
15,064
0.71%
0.71%
           
81
77
4
266
13,926
0
0
600
0
14,796
0.70%
0.70%
82
96s
4
2,200
2,026
888
0
9,600
0
14,718
0.69%
0.69%
83
66
4
266
13,652
0
0
600
0
14,522
0.69%
0.69%
84
A7o
49
133
2,542
1,470
5,066
5,100
0
14,360
0.68%
0.68%
85
85s
4
2,200
2,014
465
0
9,600
0
14,283
0.67%
0.68%
86
55
4
266
13,398
0
0
600
0
14,268
0.67%
0.67%
87
44
4
176
13,358
0
0
600
0
14,138
0.67%
0.67%
88
33
4
86
13,312
0
0
600
0
14,002
0.66%
0.66%
89
74s
4
2,155
2,011
177
0
9,600
0
13,947
0.66%
0.66%
90
22
4
0
13,262
0
0
600
0
13,866
0.65%
0.66%
           
91
A6o
49
133
2,536
870
5,061
5,100
0
13,749
0.65%
0.65%
92
63s
4
2,110
2,008
24
0
9,600
0
13,746
0.65%
0.65%
93
Q7s
49
270
2,068
1,566
4,471
5,100
0
13,524
0.64%
0.64%
94
A5o
49
133
2,557
375
5,070
5,100
0
13,284
0.63%
0.63%
95
A4o
49
88
2,548
153
5,079
5,100
0
13,017
0.61%
0.62%
96
K9o
49
178
2,143
3,222
2,324
5,100
0
13,016
0.61%
0.61%
97
A3o
49
43
2,545
28
5,089
5,100
0
12,854
0.61%
0.60%
98
Q6s
49
223
2,062
954
4,464
5,100
0
12,852
0.61%
0.60%
99
96o
4
266
2,038
910
0
9,600
0
12,818
0.60%
0.60%
100
A2o
49
0
2,548
0
5,101
5,100
0
12,798
0.60%
0.61%
           
101
Q5s
49
223
2,056
513
4,653
5,100
0
12,594
0.59%
0.59%
102
85o
4
266
2,026
481
0
9,600
0
12,377
0.58%
0.58%
103
Q4s
49
178
2,059
207
4,647
5,100
0
12,240
0.58%
0.58%
104
K8o
49
178
2,131
2,322
2,322
5,100
0
12,102
0.57%
0.57%
105
74o
4
221
2,023
187
0
9,600
0
12,035
0.57%
0.57%
106
Q3s
49
133
2,062
36
4,641
5,100
0
12,021
0.57%
0.57%
107
Q2s
49
90
2,065
0
4,636
5,100
0
11,940
0.56%
0.56%
108
63o
4
176
2,020
28
0
9,600
0
11,828
0.56%
0.56%
109
K7o
49
178
2,125
1,566
2,324
5,100
0
11,342
0.54%
0.53%
110
Q8o
49
223
2,080
2,322
1,046
5,100
0
10,820
0.51%
0.52%
           
111
K6o
49
178
2,119
954
2,327
5,100
0
10,727
0.51%
0.51%
112
J7s
49
1,259
2,053
1,566
609
5,100
0
10,636
0.50%
0.50%
113
K5o
49
178
2,113
513
2,315
5,100
0
10,268
0.48%
0.49%
114
52s
4
2,104
2,005
0
0
6,000
0
10,113
0.48%
0.47%
115
Q7o
49
223
2,068
1,566
1,046
5,100
0
10,052
0.47%
0.48%
116
K4o
49
133
2,116
207
2,318
5,100
0
9,923
0.47%
0.47%
117
K3o
49
88
2,119
36
2,320
5,100
0
9,712
0.46%
0.46%
118
K2o
49
45
2,122
0
2,321
5,100
0
9,637
0.45%
0.45%
119
T6s
49
1,304
2,032
954
0
5,100
0
9,439
0.45%
0.45%
120
Q6o
49
223
2,062
954
1,046
5,100
0
9,434
0.45%
0.44%
           
121
42s
4
1,972
2,008
0
0
5,400
0
9,384
0.44%
0.44%
122
J7o
49
268
2,059
1,566
231
5,100
0
9,273
0.44%
0.44%
123
J6s
49
315
2,047
954
609
5,100
0
9,074
0.43%
0.43%
124
Q5o
49
223
2,056
513
1,043
5,100
0
8,984
0.42%
0.42%
125
J6o
49
268
2,047
954
231
5,100
0
8,649
0.41%
0.41%
126
Q4o
49
178
2,059
207
1,043
5,100
0
8,636
0.41%
0.41%
127
J5s
49
268
2,041
513
615
5,100
0
8,586
0.41%
0.40%
128
T6o
49
313
2,038
954
0
5,100
0
8,454
0.40%
0.40%
129
Q3o
49
133
2,062
36
1,043
5,100
0
8,423
0.40%
0.40%
130
Q2o
49
90
2,065
0
1,043
5,100
0
8,347
0.39%
0.39%
           
131
J4s
49
223
2,044
207
615
5,100
0
8,238
0.39%
0.39%
132
J5o
49
268
2,041
513
231
5,100
0
8,202
0.39%
0.39%
133
52o
4
133
2,020
0
0
6,000
0
8,157
0.38%
0.38%
134
T5s
49
360
2,026
513
0
5,100
0
8,048
0.38%
0.38%
135
J3s
49
178
2,047
36
615
5,100
0
8,025
0.38%
0.38%
136
T5o
49
313
2,026
513
0
5,100
0
8,001
0.38%
0.38%
137
J2s
49
135
2,050
0
615
5,100
0
7,949
0.38%
0.37%
138
J4o
49
223
2,044
207
231
5,100
0
7,854
0.37%
0.37%
139
T4s
49
268
2,029
207
0
5,100
0
7,653
0.36%
0.36%
140
T4o
49
268
2,029
207
0
5,100
0
7,653
0.36%
0.36%
           
141
J3o
49
178
2,047
36
231
5,100
0
7,641
0.36%
0.36%
142
J2o
49
135
2,050
0
231
5,100
0
7,565
0.36%
0.35%
143
42o
4
88
2,023
0
0
5,400
0
7,515
0.35%
0.35%
144
T3s
49
223
2,032
36
0
5,100
0
7,440
0.35%
0.35%
145
T3o
49
223
2,032
36
0
5,100
0
7,440
0.35%
0.35%
146
T2s
49
180
2,035
0
0
5,100
0
7,364
0.35%
0.35%
147
T2o
49
180
2,035
0
0
5,100
0
7,364
0.35%
0.35%
148
32s
4
940
2,011
0
0
3,000
0
5,955
0.28%
0.28%
149
32o
4
43
2,026
0
0
3,000
0
5,073
0.24%
0.24%
150
95s
4
1,258
2,020
513
0
600
0
4,395
0.21%
0.21%
           
151
84s
4
1,213
2,017
207
0
600
0
4,041
0.19%
0.19%
152
73s
4
1,168
2,014
36
0
600
0
3,822
0.18%
0.18%
153
62s
4
1,123
2,011
0
0
600
0
3,738
0.18%
0.18%
154
95o
4
266
2,026
513
0
600
0
3,409
0.16%
0.16%
155
94s
4
268
2,023
207
0
600
0
3,102
0.15%
0.15%
156
94o
4
221
2,023
207
0
600
0
3,055
0.14%
0.14%
157
84o
4
221
2,023
207
0
600
0
3,055
0.14%
0.14%
158
83s
4
223
2,020
36
0
600
0
2,883
0.14%
0.14%
159
93s
4
176
2,026
36
0
600
0
2,842
0.13%
0.13%
160
93o
4
176
2,026
36
0
600
0
2,842
0.13%
0.13%
           
161
83o
4
176
2,020
36
0
600
0
2,836
0.13%
0.13%
162
73o
4
176
2,020
36
0
600
0
2,836
0.13%
0.14%
163
72s
4
178
2,017
0
0
600
0
2,799
0.13%
0.14%
164
92s
4
133
2,029
0
0
600
0
2,766
0.13%
0.13%
165
92o
4
133
2,029
0
0
600
0
2,766
0.13%
0.13%
166
82s
4
133
2,023
0
0
600
0
2,760
0.13%
0.13%
167
82o
4
133
2,023
0
0
600
0
2,760
0.13%
0.13%
168
72o
4
133
2,017
0
0
600
0
2,754
0.13%
0.13%
169
62o
4
133
2,017
0
0
600
0
2,754
0.13%
0.13%

Let me know if you have any questions or comments.
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Old 05-03-2018, 11:35 PM   #25
robert_utk
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Posts: 2,910
Re: Poker Riddle from FiveThirtyEight.com

Quote:
Originally Posted by whosnext View Post
Let me know if you have any questions or comments.
Wow. This is simply amazing. Quite a triumph! I know any complement I have to offer is shallow, since I can not appreciate all the effort you put into this. I can say for certain, that I greatly admire a logical solution versus a brute force simulation, when such a logical solution is possible.
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