Poker Mavens Statistics Question (3 Viewers)

Can you tell me how often Sicko rivers a straight or flush when I am holding a set or how often the board pairs when I am holding a flush :unsure:
The raw data has this, yes, but parsing the logic and flow behind a hand is considerably more complex.

To be clear, I'm not calling out those moments that you speak of. They are much harder to analyze correctly; that's why I focused on pocket pairs. It's a clean statistic.
 
The raw data has this, yes, but parsing the logic and flow behind a hand is considerably more complex.

To be clear, I'm not calling out those moments that you speak of. They are much harder to analyze correctly; that's why I focused on pocket pairs. It's a clean statistic.

It certainly feels like more often than not
 
I am not a maff wizard. I am also not naive about potential nefariance in online poker.

However, when I see a thread about the software being “rigged” my immediate reaction is to think our pattern sensing brains have been again fooled by randomness.

That said, I'm still curious about the math on this will so will follow the discussion.
 
1 table hand = 14 player hands
Only if all 14 players are seated at 1 table.

If it's running 2 tables with 7 players, the hand dealt at Table #1, would be counted separately from the hand dealt at Table #2, in the software, right? Then there are 2 hands, with an average of 7 players per table, for 14 player-hands over 2 table-hands.

How many seats at the final table, when winding down to 1 table? (is it still 7-handed?)

If the tournament starts 7-handed per table, and ends up 2-handed, the average players per hand for the overall tournament would be somewhere in-between.

Just running some numbers in the spreadsheet:
33,662 table-hands multiplied by 5.11 (assumed Average players dealt in per hand), would result in 172,013 Player-Hands, and the average number of pocket pairs expected would be 1 out of 17, or 10,118, which is what is above.
 
It's about 2.5x, not quite 3x. But this is the problem with (relatively) small samples. Does this player's openers represent an average result compared to other players' data? Above average? Below average? Just about every data set is going to have outliers, especially if we're talking about something that's randomized independently every iteration.

In theory, each player should get each pocket pair about an equal number of times as each other pocket pair, which clearly isn't the case here. At the extremes, we have 80 and 129, representing being dealt 22 more than 50% more often than 88. It doesn't mean the site favors 22 or disfavors 88; it's just random variance, like a coin landing on heads a bunch of times in a row. It's a little odd for one player to get an above-average result for all of his pocket pairs, but not any more odd than a player getting a below-average result for all of his pocket pairs, which of course can happen too.

Also, if your hypothesis is that the site is trying to stir up action with unequal hand distributions, shouldn't this pocket-pair data be more top-heavy? Little pairs don't stir up big action; big pairs do. And in this same stroke, I'd like to point out something you addressed right upfront, which is that Briggs has no motive for his software to produce skewed results, whereas he does have motive for it to be fairly randomized. Why should he care if his site stirs up a little more action than others, especially if using broken algorithms to achieve it could drive people away from buying his software?
This is player is one who has played in every tournament. He has the largest single player dataset, so I figured he was a good person to isolate to see if his data aligned. Actually, my point wasn't that the site was trying to use poker rules to stir up fun; it was just that we had noticed an abnormally large number of pocket pairs and when I ran the stats, it seemed to align with that theory.

I guess my point is that Briggs has no reason to do it; the license was paid. Is there a problem with the algorithm? Is there a more general problem? I dunno. I decided to not talk about "fun manufactured hands" specifically because it's much harder to analyze that data (AA vs KK; flop is AKK-A-7l; two huge, aggressive players are in it) ... yes that's fun, but much harder to prove intent of the software. Also, you can say that it's just because you see more hands that you see more scenarios like this. I'm just trying to focus on the simplest raw data to see why it seems pocket pairs come up so often...

I find it odd that your data on one player includes 9,000+ hands, while your total data pool only covers about 34,000 hands. It makes me feel like you're looking at all of the times all the different pairs have come up across all the players at the table, but still only comparing it to 33,662. If you have 33,662 hands, but 9 players were at the table for each hand, you're really looking at 302,958 openers. Most likely, your data represents somewhat fewer hands than this, since I doubt you're 9-handed all the time, but still way more than 33,662 openers.

Maybe we can test this over a large number of dummy trials. If I understand Mavens (I have a server too but haven't hosted anything yet), you can set up a table that will essentially auto-deal a ton of hands in a relatively short period, without having to involve any real players. Just make two dummy accounts, and make a cash table that's cap-limit Hold'em, with an ante equal to the cap, and duplicate IP addresses allowed. Give both dummy accounts a bazillion chips in their balance and let them buy in for the whole thing. Then just sit them down and let it roll.

In fact, I think I'll take part of my lunch break to see if I can set this up on my server. The part i don't know how to do is run queries on past hands, though, so I may swing back here and ask for help with that.
In this case a 34,000 hands is a table hand; I multiplied by the average number of players to try to normalize the data, but there's still some noise in that data. So, that's why I focused on 1 player with a good dataset.

One thing that seems interesting is that many pocket pair totals are within 2% of one another. Yes, there are outliers, but overall, it feels like there's no real runaway. The vast majority for my single player are in the 100-11x's. The vast majority site-wide are in the upper 700's. So I'm not saying it's skewing it for one pair. But when you account for all of the pocket pairs together over all hands over all players, it seems like it's just too often.
 
Here are the last 90,372 NLHE card pairs dealt on my Mavens site, along with the percentage for each combination and a "1 in" column that indicates that the hand was dealt every 1 in x hands. Pocket pairs are sorted to the top. To be clear, 90,372 is the number of card pairs, not the number of overall "hands" with a variable number of players in each. The average for pocket pairs is 1 in 221 (1/13 x 3/51); the average for non-pairs is 1 in 166 (1/13 x 4/51).

Card 1Card 2Pair?
Count​
Pct​
1 in​
22Yes
415​
0.46%​
218​
33Yes
405​
0.45%​
223​
44Yes
405​
0.45%​
223​
55Yes
447​
0.49%​
202​
66Yes
427​
0.47%​
212​
77Yes
404​
0.45%​
224​
88Yes
420​
0.46%​
215​
99Yes
376​
0.42%​
240​
TTYes
436​
0.48%​
207​
JJYes
377​
0.42%​
240​
QQYes
418​
0.46%​
216​
KKYes
409​
0.45%​
221​
AAYes
386​
0.43%​
234​
23
523​
0.58%​
173​
24
569​
0.63%​
159​
25
607​
0.67%​
149​
26
554​
0.61%​
163​
27
590​
0.65%​
153​
28
580​
0.64%​
156​
29
568​
0.63%​
159​
2A
554​
0.61%​
163​
2J
573​
0.63%​
158​
2K
543​
0.60%​
166​
2Q
578​
0.64%​
156​
2T
557​
0.62%​
162​
32
572​
0.63%​
158​
34
521​
0.58%​
173​
35
536​
0.59%​
169​
36
533​
0.59%​
170​
37
576​
0.64%​
157​
38
537​
0.59%​
168​
39
524​
0.58%​
172​
3A
529​
0.59%​
171​
3J
551​
0.61%​
164​
3K
558​
0.62%​
162​
3Q
550​
0.61%​
164​
3T
553​
0.61%​
163​
42
561​
0.62%​
161​
43
541​
0.60%​
167​
45
534​
0.59%​
169​
46
571​
0.63%​
158​
47
540​
0.60%​
167​
48
524​
0.58%​
172​
49
542​
0.60%​
167​
4A
511​
0.57%​
177​
4J
538​
0.60%​
168​
4K
550​
0.61%​
164​
4Q
531​
0.59%​
170​
4T
540​
0.60%​
167​
52
557​
0.62%​
162​
53
535​
0.59%​
169​
54
503​
0.56%​
180​
56
532​
0.59%​
170​
57
526​
0.58%​
172​
58
539​
0.60%​
168​
59
541​
0.60%​
167​
5A
566​
0.63%​
160​
5J
513​
0.57%​
176​
5K
552​
0.61%​
164​
5Q
536​
0.59%​
169​
5T
583​
0.65%​
155​
62
544​
0.60%​
166​
63
554​
0.61%​
163​
64
536​
0.59%​
169​
65
528​
0.58%​
171​
67
521​
0.58%​
173​
68
518​
0.57%​
174​
69
546​
0.60%​
166​
6A
550​
0.61%​
164​
6J
577​
0.64%​
157​
6K
577​
0.64%​
157​
6Q
505​
0.56%​
179​
6T
537​
0.59%​
168​
72
545​
0.60%​
166​
73
531​
0.59%​
170​
74
552​
0.61%​
164​
75
556​
0.62%​
163​
76
538​
0.60%​
168​
78
533​
0.59%​
170​
79
502​
0.56%​
180​
7A
544​
0.60%​
166​
7J
518​
0.57%​
174​
7K
538​
0.60%​
168​
7Q
500​
0.55%​
181​
7T
541​
0.60%​
167​
82
567​
0.63%​
159​
83
492​
0.54%​
184​
84
537​
0.59%​
168​
85
519​
0.57%​
174​
86
536​
0.59%​
169​
87
527​
0.58%​
171​
89
557​
0.62%​
162​
8A
564​
0.62%​
160​
8J
559​
0.62%​
162​
8K
544​
0.60%​
166​
8Q
562​
0.62%​
161​
8T
554​
0.61%​
163​
92
549​
0.61%​
165​
93
491​
0.54%​
184​
94
547​
0.61%​
165​
95
519​
0.57%​
174​
96
549​
0.61%​
165​
97
543​
0.60%​
166​
98
548​
0.61%​
165​
9A
611​
0.68%​
148​
9J
583​
0.65%​
155​
9K
509​
0.56%​
178​
9Q
561​
0.62%​
161​
9T
560​
0.62%​
161​
A2
542​
0.60%​
167​
A3
533​
0.59%​
170​
A4
558​
0.62%​
162​
A5
550​
0.61%​
164​
A6
569​
0.63%​
159​
A7
561​
0.62%​
161​
A8
553​
0.61%​
163​
A9
552​
0.61%​
164​
AJ
538​
0.60%​
168​
AK
573​
0.63%​
158​
AQ
536​
0.59%​
169​
AT
549​
0.61%​
165​
J2
570​
0.63%​
159​
J3
571​
0.63%​
158​
J4
537​
0.59%​
168​
J5
555​
0.61%​
163​
J6
549​
0.61%​
165​
J7
554​
0.61%​
163​
J8
531​
0.59%​
170​
J9
556​
0.62%​
163​
JA
560​
0.62%​
161​
JK
549​
0.61%​
165​
JQ
546​
0.60%​
166​
JT
586​
0.65%​
154​
K2
533​
0.59%​
170​
K3
553​
0.61%​
163​
K4
516​
0.57%​
175​
K5
522​
0.58%​
173​
K6
545​
0.60%​
166​
K7
542​
0.60%​
167​
K8
541​
0.60%​
167​
K9
516​
0.57%​
175​
KA
513​
0.57%​
176​
KJ
504​
0.56%​
179​
KQ
583​
0.65%​
155​
KT
565​
0.63%​
160​
Q2
577​
0.64%​
157​
Q3
535​
0.59%​
169​
Q4
540​
0.60%​
167​
Q5
501​
0.55%​
180​
Q6
542​
0.60%​
167​
Q7
528​
0.58%​
171​
Q8
545​
0.60%​
166​
Q9
514​
0.57%​
176​
QA
555​
0.61%​
163​
QJ
525​
0.58%​
172​
QK
511​
0.57%​
177​
QT
561​
0.62%​
161​
T2
553​
0.61%​
163​
T3
497​
0.55%​
182​
T4
548​
0.61%​
165​
T5
539​
0.60%​
168​
T6
562​
0.62%​
161​
T7
616​
0.68%​
147​
T8
549​
0.61%​
165​
T9
542​
0.60%​
167​
TA
532​
0.59%​
170​
TJ
581​
0.64%​
156​
TK
561​
0.62%​
161​
TQ
541​
0.60%​
167​
 
I am not a maff wizard. I am also not naive about potential nefariance in online poker.

However, when I see a thread about the software being “rigged” my immediate reaction is to think our pattern sensing brains have been again fooled by randomness.

That said, I'm still curious about the math on this will so will follow the discussion.
As I said in my OP; I'm the last person to create this thread. I, too, have stood firm with "it's your imagination and because you play so many more hands online than in person". But it's when I ran the stats that I got more curious...

Only if all 14 players are seated at 1 table.

If it's running 2 tables with 7 players, the hand dealt at Table #1, would be counted separately from the hand dealt at Table #2, in the software, right? Then there are 2 hands, with an average of 7 players per table, for 14 player-hands over 2 table-hands.

How many seats at the final table, when winding down to 1 table? (is it still 7-handed?)

If the tournament starts 7-handed per table, and ends up 2-handed, the average players per hand for the overall tournament would be somewhere in-between.

Just running some numbers in the spreadsheet:
33,662 table-hands multiplied by 5.11 (assumed Average players dealt in per hand), would result in 172,013 Player-Hands, and the average number of pocket pairs expected would be 1 out of 17, or 10,118, which is what is above.
Yes, I know it's not precise turning 2-table tournament hands into player hands. That's why I tried to limit it to a single player as well. If we forget "tournament hands" and focus on player hands, then we get something like this. I picked 2 random players; their percentages for pocket pairs are just way too high!

Player 1 (24,717)
105
111
98
108
90
87
94
79
106
88
94
90
106
Player 2 (15,691)
89
76
77
80
79
88
91
77
82
82
72
79
73

Edit: whoops; pasted wrong total hands :)
 
Here are the last 90,372 NLHE card pairs dealt on my Mavens site, along with the percentage for each combination and a "1 in" column that indicates that the hand was dealt every 1 in x hands. Pocket pairs are sorted to the top. To be clear, 90,372 is the number of card pairs, not the number of overall "hands" with a variable number of players in each. The average for pocket pairs is 1 in 221 (1/13 x 3/51); the average for non-pairs is 1 in 166 (1/13 x 4/51).

Card 1Card 2Pair?
Count​
Pct​
1 in​
22Yes
415​
0.46%​
218​
33Yes
405​
0.45%​
223​
44Yes
405​
0.45%​
223​
55Yes
447​
0.49%​
202​
66Yes
427​
0.47%​
212​
77Yes
404​
0.45%​
224​
88Yes
420​
0.46%​
215​
99Yes
376​
0.42%​
240​
TTYes
436​
0.48%​
207​
JJYes
377​
0.42%​
240​
QQYes
418​
0.46%​
216​
KKYes
409​
0.45%​
221​
AAYes
386​
0.43%​
234​
23
523​
0.58%​
173​
24
569​
0.63%​
159​
25
607​
0.67%​
149​
26
554​
0.61%​
163​
27
590​
0.65%​
153​
28
580​
0.64%​
156​
29
568​
0.63%​
159​
2A
554​
0.61%​
163​
2J
573​
0.63%​
158​
2K
543​
0.60%​
166​
2Q
578​
0.64%​
156​
2T
557​
0.62%​
162​
32
572​
0.63%​
158​
34
521​
0.58%​
173​
35
536​
0.59%​
169​
36
533​
0.59%​
170​
37
576​
0.64%​
157​
38
537​
0.59%​
168​
39
524​
0.58%​
172​
3A
529​
0.59%​
171​
3J
551​
0.61%​
164​
3K
558​
0.62%​
162​
3Q
550​
0.61%​
164​
3T
553​
0.61%​
163​
42
561​
0.62%​
161​
43
541​
0.60%​
167​
45
534​
0.59%​
169​
46
571​
0.63%​
158​
47
540​
0.60%​
167​
48
524​
0.58%​
172​
49
542​
0.60%​
167​
4A
511​
0.57%​
177​
4J
538​
0.60%​
168​
4K
550​
0.61%​
164​
4Q
531​
0.59%​
170​
4T
540​
0.60%​
167​
52
557​
0.62%​
162​
53
535​
0.59%​
169​
54
503​
0.56%​
180​
56
532​
0.59%​
170​
57
526​
0.58%​
172​
58
539​
0.60%​
168​
59
541​
0.60%​
167​
5A
566​
0.63%​
160​
5J
513​
0.57%​
176​
5K
552​
0.61%​
164​
5Q
536​
0.59%​
169​
5T
583​
0.65%​
155​
62
544​
0.60%​
166​
63
554​
0.61%​
163​
64
536​
0.59%​
169​
65
528​
0.58%​
171​
67
521​
0.58%​
173​
68
518​
0.57%​
174​
69
546​
0.60%​
166​
6A
550​
0.61%​
164​
6J
577​
0.64%​
157​
6K
577​
0.64%​
157​
6Q
505​
0.56%​
179​
6T
537​
0.59%​
168​
72
545​
0.60%​
166​
73
531​
0.59%​
170​
74
552​
0.61%​
164​
75
556​
0.62%​
163​
76
538​
0.60%​
168​
78
533​
0.59%​
170​
79
502​
0.56%​
180​
7A
544​
0.60%​
166​
7J
518​
0.57%​
174​
7K
538​
0.60%​
168​
7Q
500​
0.55%​
181​
7T
541​
0.60%​
167​
82
567​
0.63%​
159​
83
492​
0.54%​
184​
84
537​
0.59%​
168​
85
519​
0.57%​
174​
86
536​
0.59%​
169​
87
527​
0.58%​
171​
89
557​
0.62%​
162​
8A
564​
0.62%​
160​
8J
559​
0.62%​
162​
8K
544​
0.60%​
166​
8Q
562​
0.62%​
161​
8T
554​
0.61%​
163​
92
549​
0.61%​
165​
93
491​
0.54%​
184​
94
547​
0.61%​
165​
95
519​
0.57%​
174​
96
549​
0.61%​
165​
97
543​
0.60%​
166​
98
548​
0.61%​
165​
9A
611​
0.68%​
148​
9J
583​
0.65%​
155​
9K
509​
0.56%​
178​
9Q
561​
0.62%​
161​
9T
560​
0.62%​
161​
A2
542​
0.60%​
167​
A3
533​
0.59%​
170​
A4
558​
0.62%​
162​
A5
550​
0.61%​
164​
A6
569​
0.63%​
159​
A7
561​
0.62%​
161​
A8
553​
0.61%​
163​
A9
552​
0.61%​
164​
AJ
538​
0.60%​
168​
AK
573​
0.63%​
158​
AQ
536​
0.59%​
169​
AT
549​
0.61%​
165​
J2
570​
0.63%​
159​
J3
571​
0.63%​
158​
J4
537​
0.59%​
168​
J5
555​
0.61%​
163​
J6
549​
0.61%​
165​
J7
554​
0.61%​
163​
J8
531​
0.59%​
170​
J9
556​
0.62%​
163​
JA
560​
0.62%​
161​
JK
549​
0.61%​
165​
JQ
546​
0.60%​
166​
JT
586​
0.65%​
154​
K2
533​
0.59%​
170​
K3
553​
0.61%​
163​
K4
516​
0.57%​
175​
K5
522​
0.58%​
173​
K6
545​
0.60%​
166​
K7
542​
0.60%​
167​
K8
541​
0.60%​
167​
K9
516​
0.57%​
175​
KA
513​
0.57%​
176​
KJ
504​
0.56%​
179​
KQ
583​
0.65%​
155​
KT
565​
0.63%​
160​
Q2
577​
0.64%​
157​
Q3
535​
0.59%​
169​
Q4
540​
0.60%​
167​
Q5
501​
0.55%​
180​
Q6
542​
0.60%​
167​
Q7
528​
0.58%​
171​
Q8
545​
0.60%​
166​
Q9
514​
0.57%​
176​
QA
555​
0.61%​
163​
QJ
525​
0.58%​
172​
QK
511​
0.57%​
177​
QT
561​
0.62%​
161​
T2
553​
0.61%​
163​
T3
497​
0.55%​
182​
T4
548​
0.61%​
165​
T5
539​
0.60%​
168​
T6
562​
0.62%​
161​
T7
616​
0.68%​
147​
T8
549​
0.61%​
165​
T9
542​
0.60%​
167​
TA
532​
0.59%​
170​
TJ
581​
0.64%​
156​
TK
561​
0.62%​
161​
TQ
541​
0.60%​
167​
Now that's what I call data!
 
I ran some numbers on my own hand histories, and found the following:

11879 total cash game hands. I only found 30 instances of myself with AA, which multiplied by 221 = 6630. I am usually only in the cash games about half the time they run, so that seems about right to me.
 
Here are the last 90,372 NLHE card pairs dealt on my Mavens site, along with the percentage for each combination and a "1 in" column that indicates that the hand was dealt every 1 in x hands. Pocket pairs are sorted to the top. To be clear, 90,372 is the number of card pairs, not the number of overall "hands" with a variable number of players in each. The average for pocket pairs is 1 in 221 (1/13 x 3/51); the average for non-pairs is 1 in 166 (1/13 x 4/51).
Let me try running a similar query on mine ... one moment while I jack-in ...
 

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Something doesn't appear right in the data.

There were 10,118 pocket pairs dealt to all players in all hands. Nothing to question here.

You picked 2 players out of 14 players, and their pocket-pairs totals are 1,425 + 1,045 = total of 2,470. That would be close to 25% of all the pocket pairs dealt in all hands of 10,118. (Or do these 2 often make the final table and cash, and would play 25% of the total hands?)

Somehow I think the data for individual players is not right, or the query for individual players is pulling other unintended data.

1614018070442.png
 
OK, so my site has seen 156,351 unique player hands. If I do a very simply summary here, I get this. This is taking into account some extra games which were not in my original set (play test games for checking cards/chips ... but still, they were hands dealt)

HandsOccurrences1-in-x Hands
Total: 156351
AA827189
KK772203
QQ811193
JJ803195
TT799196
99820191
88783200
77777201
66826189
55739212
44826189
33805194
22830188

So that's still more often, right? Maybe not as high as my original numbers, but it's happening more often than it should...
 
Last edited:
As I said in my OP; I'm the last person to create this thread. I, too, have stood firm with "it's your imagination and because you play so many more hands online than in person". But it's when I ran the stats that I got more curious...


Yes, I know it's not precise turning 2-table tournament hands into player hands. That's why I tried to limit it to a single player as well. If we forget "tournament hands" and focus on player hands, then we get something like this. I picked 2 random players; their percentages for pocket pairs are just way too high!

Player 1 (24,717)
105
111
98
108
90
87
94
79
106
88
94
90
106
Player 2 (15,691)
89
76
77
80
79
88
91
77
82
82
72
79
73

Edit: whoops; pasted wrong total hands :)
This data falls much more closely to the theoretical probability of being dealt a pair of whatever.

Player 1 falls a little bit under the curve; a "perfect" data set would have him catching each pocket pair about 112 times out of 24,717 openers. Player 2 falls a little bit above the curve; his "perfect" data would be about 71 of each pocket pair.

The previous sample player you provided has apparently run pretty hot (with openers, anyway) over the tournaments that form your data.
 
You picked 2 players out of 14 players, and their pocket-pairs totals are 1,425 + 1,045 = total of 2,470. That would be close to 25% of all the pocket pairs dealt in all hands of 10,118. (Or do these 2 often make the final table and cash, and would play 25% of the total hands?)

Somehow I think the data for individual players is not right, or the query for individual players is pulling other unintended data.
I chose players who play way more than any other player and they are the most often to cash and have the highest win %. I posted some raw-er stats up above. I guess it makes the data look better (not 5x as often, but still seems too often)...
 
OK, so my site has seen 156,351 unique player hands. If I do a very simply summary here, I get this. This is taking into account some extra games which were not in my original set (play test games for checking cards/chips ... but still, they were hands dealt)

156351827189.0580411
772202.5272021
811192.7879162
803194.7085928
799195.6833542
820190.6719512
783199.6819923
777201.2239382
826189.2869249
739211.5710419
826189.2869249
805194.2248447
830188.3746988

So that's still more often, right? Maybe not as high as my original numbers, but it's happening more often than it should...
I mean, yeah, this is still more often than 1 in 221 for each individual pair, but it's just barely off of that theoretical average, well within any reasonable standard of variability. This is worlds different from the first chart you provided.
 
I mean, yeah, this is still more often than 1 in 221 for each individual pair, but it's just barely off of that theoretical average, well within any reasonable standard of variability. This is worlds different from the first chart you provided.
Yup, I agree; it seems as though I just needed to take into account all hands not table hands. Still more often (for every pocket pair) ... but not 5x
 
I'm running the bot game that I mentioned above just to see, but at the rate it's going, it's only going to deal out about 6,720 hands a day (or 13,440 "player hands" heads-up) which doesn't help a whole lot here. If I can find a way to make it work 9- or 10-handed so that we can get more data, I'll give it a shot. But it seems like we're reaching the point where we realize that the massive discrepancies were likely from data-analysis error and not actually players getting pairs disproportionately often.

If any of you care to check out the bot test, go to the private site Rungood Lounge on Mavens, PW dramaha, cash table RNG Test.
 
I'm running the bot game that I mentioned above just to see, but at the rate it's going, it's only going to deal out about 6,720 hands a day (or 13,440 "player hands" heads-up) which doesn't help a whole lot here. If I can find a way to make it work 9- or 10-handed so that we can get more data, I'll give it a shot. But it seems like we're reaching the point where we realize that the massive discrepancies were likely from data-analysis error and not actually players getting pairs disproportionately often.
Yes, the massive discrepancy was due to analyzing the wrong hand types; but when it switches over to just # of player hands vs # of pocket pairs, it's still too often. Maybe it's within some tolerance, but not 1 of my pocket pairs is above 1-in-221 ... most are in the 180s...
 
Yes, the massive discrepancy was due to analyzing the wrong hand types; but when it switches over to just # of player hands vs # of pocket pairs, it's still too often. Maybe it's within some tolerance, but not 1 of my pocket pairs is above 1-in-221 ... most are in the 180s...
What you're seeing in your data isn't really "too often." It's just more often than the theoretical average, but it's not so far off that you'd call it abnormal, especially given the modest sample size.

It's a little odd that all of your pocket pairs are coming in more often than 1 in 221; you'd expect there to be a random smattering of some above and some below that line. But again, not so far off that it should raise any alarms.
 
It's a little odd that all of your pocket pairs are coming in more often than 1 in 221; you'd expect there to be a random smattering of some above and some below that line. But again, not so far off that it should raise any alarms.
Yup, this was my point. After changing the analysis, we're not beyond the realm of theoretical probability for happening too often (at least I don't think we are). But the fact that they ALL are below the 1-in-221 is odd. I guess that's the data that caused myself (and my players) to say "Aren't these freaking PPs coming up too often?"

My OP is updated.
 
Here are the last 90,372 NLHE card pairs dealt on my Mavens site, along with the percentage for each combination and a "1 in" column that indicates that the hand was dealt every 1 in x hands. Pocket pairs are sorted to the top. To be clear, 90,372 is the number of card pairs, not the number of overall "hands" with a variable number of players in each. The average for pocket pairs is 1 in 221 (1/13 x 3/51); the average for non-pairs is 1 in 166 (1/13 x 4/51).

This is the analysis I would have done.

Based on the second data @BearMetal presented it looks like they are getting pairs about 10% more often than expected, but I wouldn't take that alone as a sign something's wrong, I would expect that sort of variance. When we were talking twice-three times as often, I was concerned.
 
This is the analysis I would have done.

Based on the second data @BearMetal presented it looks like they are getting pairs about 10% more often than expected, but I wouldn't take that alone as a sign something's wrong, I would expect that sort of variance. When we were talking twice-three times as often, I was concerned.
Yup, that's why I brought it up here. Wanted to get to the bottom of it. Something seemed wrong. Glad we got it straightened out. Still odd they are all happening 10-15% more often ...
 
Yup, that's why I brought it up here. Wanted to get to the bottom of it. Something seemed wrong. Glad we got it straightened out. Still odd they are all happening 10-15% more often ...
I suspect if you do the analysis exactly like the other one with all hand combos the stats might be a little different. I might be wrong on that but that's my gut.
 
I suspect if you do the analysis exactly like the other one with all hand combos the stats might be a little different. I might be wrong on that but that's my gut.
I thought that's what I did. I went back and compared the total hands dealt to all players to the number of times we saw pocket pairs.
 

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