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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
CallMeVernon
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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
CallMeVernon
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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
whosnext
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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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