BBO Robot Hands

[tx10s]

When I posted this, the data columns lined up nicely. When I went back to add a comment, I see that the data does not present well. In each row, the first item is the date, the second is the tournament number. The third is finesses onside for the players, the fourth is finesses offside for the player, the fifth is finesses onside for the robots, the sixth is finesses offside for the robot, the seventh is the sum of finesses onside for the players and offside for the robots (in other words the finesses that favor the player) and the eights data point is the sum of finesses offside for the player and onside for the robots.

Sorry, I had blanks instead of zeros, here is the same data again with the zeros, although still not well presented. I have tried several methods without success, so I will see if I can get Google spreadsheets to work.

 

		Finesses					
	Player		Robots		Total	
Date	Tourn	Win	Lose	Win	Lose	Win	Lose
31-Dec	5357	4	6	1	0	4	7
1-Jan	9466	5	6	2	3	8	8
1-Jan	9705	6	7	0	1	7	7
1-Jan	864	2	3	0	0	2	3
2-Jan	7203	4	6	0	1	5	6
3-Jan	3698	3	9	3	1	4	12
4-Jan	9091	7	10	1	2	9	11
4-Jan	92	6	5	4	3	9	9
5-Jan	5308	4	5	7	1	5	12
5-Jan	6810	4	12	1	1	5	13
6-Jan	1798	4	1	0	1	5	1
6-Jan	3864	3	7	1	0	3	8
6-Jan	4027	7	1	1	0	7	2
8-Jan	3839	7	12	0	0	7	12
8-Jan	4589	3	11	2	2	5	13
10-Jan	1077	2	7	2	0	2	9
11-Jan	5532	3	9	2	1	4	11
11-Jan	6086	4	3	1	0	4	4
12-Jan	1855	4	5	5	1	5	10
12-Jan	2753	3	7	4	2	5	11
						
Totals	85	132	37	20	105	169

 

Please remember that totals are Wins: Player wins plus robot losses and Losses: Player losses plus robot wins

[barmar]

Use the "Code" style to post formatted tables. It's in the Other Styles menu.

[tx10s]

Since I don't see how to post the spreadsheet, and my dataset isn't all that big, I'll copy and paste it here, although I'd still appreciate it if someone explained how to post a file. Well, I can't get it to format nice, and I can't find the recent post about how to make nice columns, so I'll try again tomorrow...

When I see how to make nice columns, I will try to post more of my data. I am surprised that this forum has no obvious way to post a spreadsheet, or down load any file for that matter. I am hoping there is a way, and one of you experts can tell me.

[tx10s]

Sorry, I had blanks instead of zeros, here is the same data again with the zeros, although still not well presented. I have tried several methods without success, so I will see if I can get Google spreadsheets to work.

 

		Finesses					
	Player		Robots		Total	
Date	Tourn	Win	Lose	Win	Lose	Win	Lose
31-Dec	5357	4	6	1	0	4	7
1-Jan	9466	5	6	2	3	8	8
1-Jan	9705	6	7	0	1	7	7
1-Jan	864	2	3	0	0	2	3
2-Jan	7203	4	6	0	1	5	6
3-Jan	3698	3	9	3	1	4	12
4-Jan	9091	7	10	1	2	9	11
4-Jan	92	6	5	4	3	9	9
5-Jan	5308	4	5	7	1	5	12
5-Jan	6810	4	12	1	1	5	13
6-Jan	1798	4	1	0	1	5	1
6-Jan	3864	3	7	1	0	3	8
6-Jan	4027	7	1	1	0	7	2
8-Jan	3839	7	12	0	0	7	12
8-Jan	4589	3	11	2	2	5	13
10-Jan	1077	2	7	2	0	2	9
11-Jan	5532	3	9	2	1	4	11
11-Jan	6086	4	3	1	0	4	4
12-Jan	1855	4	5	5	1	5	10
12-Jan	2753	3	7	4	2	5	11
						
Totals	85	132	37	20	105	169

 

Please remember that totals are Wins: Player wins plus robot losses and Losses: Player losses plus robot wins

Thank you to whoever fixed my post

[helene_t]

If there's a tie for the best hand among the other seats, we pick one of them at random.

 

I don't see how this could result in a finesse bias, though.

If you were more likely to swap with East than with West when EW are tied for best hand, it would produce losing finesses for declarer.

[barmar]

In the "Robot finesse" column, are these cases where the robot took a finesse and it won or lost, or are they cases where the robot COULD have taken a finesse, and it would have won or lost.

 

With AQxx opposite a singleton, and sufficient transportation and ruffing power, the robots often try to ruff out the K rather than just take the finesse, especially if the bidding indicates strength in the hand behind the AQ. So it probably takes fewer finesses than you do, and takes them in cases where they're more likely to succeed.

 

Basically, I think what you're seeing is that the robots are better at calculating the odds and using them to guide their play. This is the kind of thing computers are good at.

[tx10s]

Use the "Code" style to post formatted tables. It's in the Other Styles menu.

		Finesses					
	Player		Robots		Total	
Date	Tourn	Win	Lose	Win	Lose	Win	Lose
29-Oct	254	5	8	0	0	5	8
28-Oct	4033	1	6	1	2	3	7
28-Oct	2207	7	10	0	0	7	10
27-Oct	7233	4	8	1	0	4	9
26-Oct	9970	7	2	2	1	8	4
25-Oct	5406	8	8	1	2	10	9
24-Oct	9445	3	9	2	1	4	11
24-Oct	8268	2	9	0	0	2	9
23-Oct	2680	7	5	1	1	8	6
23-Oct	2038	1	3	0	0	1	3
22-Oct	7000	4	4	1	0	4	5
22-Oct	5668	2	5	2	3	5	7
21-Oct	718	6	5	2		6	7
21-Oct	524	4	9	2	4	8	11
20-Oct	3821	4	11	0	0	4	11
19-Oct	9454	5	6	0	0	5	6
19-Oct	7468	4	4	1	1	5	5
18-Oct	2709		6	0	1	1	6
18-Oct	1277	5	7	2	2	7	9
17-Oct	7281	4	6	0	0	4	6
						
Totals	83	131	18	18	101	149

Thanks for the tip, it works well.  Attached is another set of data in my finesses evaluation, dates are 2014

[tx10s]

		Finesses					
	Player		Robots		Total	
Date	Tourn	Win	Lose	Win	Lose	Win	Lose
29-Oct	254	5	8	0	0	5	8
28-Oct	4033	1	6	1	2	3	7
28-Oct	2207	7	10	0	0	7	10
27-Oct	7233	4	8	1	0	4	9
26-Oct	9970	7	2	2	1	8	4
25-Oct	5406	8	8	1	2	10	9
24-Oct	9445	3	9	2	1	4	11
24-Oct	8268	2	9	0	0	2	9
23-Oct	2680	7	5	1	1	8	6
23-Oct	2038	1	3	0	0	1	3
22-Oct	7000	4	4	1	0	4	5
22-Oct	5668	2	5	2	3	5	7
21-Oct	718	6	5	2		6	7
21-Oct	524	4	9	2	4	8	11
20-Oct	3821	4	11	0	0	4	11
19-Oct	9454	5	6	0	0	5	6
19-Oct	7468	4	4	1	1	5	5
18-Oct	2709		6	0	1	1	6
18-Oct	1277	5	7	2	2	7	9
17-Oct	7281	4	6	0	0	4	6
						
Totals	83	131	18	18	101	149

Thanks for the tip, it works well.  Attached is another set of data in my finesses evaluation, dates are 2014

 

Here is another set, please excuse my using a reply to post the data, but for some reason, the Code style will not copy a full spreadsheet in the Fast Replay section:

Finesses					
Player		Robots		Total	
Tourn	Win	Lose	Win	Lose	Win	Lose
7100	7	5	1	0	7	6
1024	6	6	1	3	9	7
9941	3	9	2	0	3	11
1065	4	4	4	3	7	8
8485	6	10	0	0	6	10
8890	5	4	2	1	6	6
5981	4	5	3	1	5	8
56	3	7	3	1	4	10
564	7	5	2	0	7	7
2156	3	6	2	0	3	8
6396	4	5	2	0	4	7
6703	5	7	0	0	5	7
8371	2	7	1	0	2	8
8482	4	6	0	0	4	6
1936	2	4	2	3	5	6
2075	3	8	0	1	4	8
3540	4	9	1	1	5	10
6698	6	9	1	2	8	10
2715	3	8	0	0	3	8
2936	8	7	2	2	10	9
					
Totals	89	131	29	18	107	160

[barmar]

Try clicking on Use Full Editor.

[tx10s]

In the "Robot finesse" column, are these cases where the robot took a finesse and it won or lost, or are they cases where the robot COULD have taken a finesse, and it would have won or lost.

 

With AQxx opposite a singleton, and sufficient transportation and ruffing power, the robots often try to ruff out the K rather than just take the finesse, especially if the bidding indicates strength in the hand behind the AQ. So it probably takes fewer finesses than you do, and takes them in cases where they're more likely to succeed.

 

Basically, I think what you're seeing is that the robots are better at calculating the odds and using them to guide their play. This is the kind of thing computers are good at.

An interesting point, but my finesse calculations were only based on potential finesses, I did not record which finesses were actually taken, as my analysis was done after the hands were played by just looking at the card distribution.

[1eyedjack]

I am missing something but I don't understand why (excluding total columns) there are 4 columns when I would expect 2.

 

Every finesse that is working for one side is losing for the other. Say you (human) are South, North has AQ and East has K. This is a working finesse for the robot, failing for human.

 

So I would expect this to count +1 in both the "player lose" and "robot win" columns. And I would expect this double entry to balance for each example, so that the total for the player lose column should never depart from the same total for the robot win column. But in your spreadsheet these totals do differ. What's that all about?

 

 

 

[barmar]

If you were more likely to swap with East than with West when EW are tied for best hand, it would produce losing finesses for declarer.

There's a very tiny bias. We use rand()%3 to choose a seat to swap with (0 = W, 1 = N, 2 = E), and rand() returns a number from 0 to 2^31-1. There's one more value whose modulus is 0 than 1 or 2, so there's a 0.00000014% bias towards West. But this has to be multiplied by the probability that EW are tied for best hand in the first place; this can't be any more than 1 in 16, so we're talking about only 0.000000008%.

 

For this to become noticeable you'd have to examine billions of hands, not the few hundred that tx10 and bbradley62 looked at.

[barmar]

For this to become noticeable you'd have to examine billions of hands, not the few hundred that tx10 and bbradley62 looked at.

BTW, we currently generate 10-15 million hands a year. So the total number of hands produced in the entire history of BBO is probably around 100 million. Nowhere close to making that bias noticeable even if we examined every hand (we can't actually do that -- I think our hand archives only go back to around 2007).

[Bbradley62]

Every finesse that is working for one side is losing for the other. Say you (human) are South, North has AQ and East has K. This is a working finesse for the robot, failing for human.

I don't know OP's methodology, but I over-simplified by only looking at it from the point of view of declarer. Suppose:

[hv=pc=n&s=sat63hakqd432c432&w=sk94h5432dakqc765&n=sq852h876d765ckq8&e=sj7hjt9djt98cajt9]399|300[/hv]If NS declares, the spade finesse (hoping to find K in front of the Q) is successful. If EW declares, the spade finesse (hoping to find A in front of the K) is also successful.

[hrothgar]

If you read the post, the sampling was done on hands that were already dealt. The fact that the finesse split ranged between 38.9% and 41% for sets of twenty tournaments averaging 281 finesses/ open endings with an overall average of 40.0% I felt was significant enough

 

Here's the rub...

 

People are very good at finding patterns in data. If you spend long enough staring at a random set of hands, odds are your going to find something weird about it.

 

On this set of hands, you're upset because the finesses don't work

On that set of hands, you're annoyed because E/W gets more HCPs than North South

On some other set of hands, the clubs never break well

 

It is for this reason that you start by specifying a hypothesis, and then test your hypothesis using a completely different data set.

 

This notion is fundamental to any kind of serious analysis...

[uday]

Being bored, and trying to postpone walking the dog in 34-degree weather, I looked at the set of deals in our live database

 

 

There are about 137 K deals in there. Mostly very recent deals, a handful of manually uploaded goulash deals, no doubt, but almost exclusively hands dealt by our dealer, including many 'best hand' deals

 

 

When South had the AQ of spades ( 6774 times ), west had the king 33% of the time and north 33% ( rounding to integers )

 

When West had the AQ of diamonds ( 4998 times ), north had the K 32% of the time, S had the king 36% of the time

 

When North had the AQ of hearts ( 5054 times), west had the K 31% of the time, south 37%

 

 

I'm fairly confident it is only bec of best hand that S has the K more often than the others but I don't really see a problem here. If I had to, I could find all the original poster's hands from our archives and do the same thing on those but that would be a bit of a chore.

 

Anyway - it was an interesting exercise, not that I know if i demonstrated anything

[tx10s]

I am missing something but I don't understand why (excluding total columns) there are 4 columns when I would expect 2.

 

Every finesse that is working for one side is losing for the other. Say you (human) are South, North has AQ and East has K. This is a working finesse for the robot, failing for human.

 

So I would expect this to count +1 in both the "player lose" and "robot win" columns. And I would expect this double entry to balance for each example, so that the total for the player lose column should never depart from the same total for the robot win column. But in your spreadsheet these totals do differ. What's that all about?

The last two columns cover the total. The first set of 4 columns just break down the finesses by hands played by the player and hands played by the robots, thus the last two columns are sums as you describe.

[tx10s]

Here's the rub...

 

People are very good at finding patterns in data. If you spend long enough staring at a random set of hands, odds are your going to find something weird about it.

 

On this set of hands, you're upset because the finesses don't work

On that set of hands, you're annoyed because E/W gets more HCPs than North South

On some other set of hands, the clubs never break well

 

It is for this reason that you start by specifying a hypothesis, and then test your hypothesis using a completely different data set.

 

This notion is fundamental to any kind of serious analysis...

I have a hard time taking your comments seriously. I simply looked at various randomly selected tournaments to determine how the finesses were splitting. They were looked at after the fact without recording which actual finesses were taken. I may be only a Bronze Life Master, but I can guarantee you that I can identify finesses and which direction they favor. The fact that the split so consistently averaged 40% over so many sets of tournament is a very strong indication of built in bias. The fact that the split "miraculously" changed to 50% after my posting leaves a strong indication that someone found the bias and "fixed" it. I suspect it was an input variable to the dealing program, but obviously I could never prove it.

[barmar]

The fact that the split "miraculously" changed to 50% after my posting leaves a strong indication that someone found the bias and "fixed" it. I suspect it was an input variable to the dealing program, but obviously I could never prove it.

There have been no recent changes to the dealing code. The last change was when we added the best-hand feature a few years ago.

 

What you're probably seeing is "clumping". Random values are not always evenly distributed in any particular sample, there are often bursts of one type or another. What's important is that you can never predict them.

 

http://en.wikipedia.org/wiki/Poisson_clumping

 

This is a common misconception that many people have about random data. If you look at a picture of lots of dots, and it looks uniformly fuzzy, it's probably NOT random. A random picture would have several concentrations of lightness and darkness.

[hrothgar]

I have a hard time taking your comments seriously. I simply looked at various randomly selected tournaments to determine how the finesses were splitting. They were looked at after the fact without recording which actual finesses were taken. I may be only a Bronze Life Master, but I can guarantee you that I can identify finesses and which direction they favor. The fact that the split so consistently averaged 40% over so many sets of tournament is a very strong indication of built in bias. The fact that the split "miraculously" changed to 50% after my posting leaves a strong indication that someone found the bias and "fixed" it. I suspect it was an input variable to the dealing program, but obviously I could never prove it.

 

I'll see your "bronze life master" and raise you an MIT graduate degree...

 

The only thing that your posting are doing is demonstrating that you don't even understand what issues are being discussed, let alone being able to make any meaningful contribution. The insinuation that the dealing code was "fixed" based on your expose would be hysterical, if it weren't so sad...

[barmar]

I'll see your "bronze life master" and raise you an MIT graduate degree...

Not to mention the years you spent developing statistical software at The Mathworks.

[uday]

Richard's credentials are impeccable but ... kumbaya isn't just a river in Egypt, as they don't say.

 

I've seen nuttier bugs than this. I'm convinced there is no bug here, but perhaps I need to demonstrate this with hard data from the archives.

 

When we released our html-based rewrite of our 'Just play bridge' game, someone complained that we kept redealing the same hands ( and that other than that, the game was fun). Took us a while to realize he was serious, but turns out that under certain cache settings, some players were getting the same hands redealt a few times.

 

uday

[hrothgar]

The fact that the split so consistently averaged 40% over so many sets of tournament is a very strong indication of built in bias. The fact that the split "miraculously" changed to 50% after my posting leaves a strong indication that someone found the bias and "fixed" it. I suspect it was an input variable to the dealing program, but obviously I could never prove it.

 

Here's a less conspiracy minded explanation regarding what happened

 

1. You noticed a pattern in a given set of hands

2. The pattern was real, but it happened by chance

3. Accordingly, when you examined a larger set of hands the pattern did not repeat itself

 

If you look back across the thread, you'll note that all my posts emphasize the need to test your hypothesis using a fresh set of hands.

[barmar]

To make my point about clustering more concrete, suppose you threw a coin 1000 times, and the result is like this:

 

TTTTTTT H T H TTTTT HH T H TTT H T HH T H T H TT HH T HHHH TTTT H TT HH TTTT H T H TT H TTTTT HHHH TT HHHHH T HHHH TT HHH TTTT H T H TT H TT H T HH
H TT H T H T HH TT H T H TTTT H T H TTTT H T HH TT HHH T H TT HH T HH T HHHH TT HH T H TTTT H T H T H TT H TTT H TTT HHHHHH T H TTTTT HHH T H T H T HH T H TT
HHHHHHH TTT HHHH T HHH TTTT HHH T HH T HHH T HH TT HH T H TTT H TTT H T HH TT H T H T H TT H T H T HHHH T H T H T H TT HH TT H T HHHHH T H T H TTT H TTT H
H TTTTT HH TTTT H TT H T HHH TTT HHH T HHHHH TT HHH TT HH T H T HH T HHHH T HHH TTTTTTT H T HHH TTT HH T H TTTTTTT H TTT HHH TTTTTTTT HH T H
H T H T HHH TT H TTT HHHH TT H T HHHH T HHH T HH TTTTT H TTT H T H TTT H T HHH TTT H TTTTTTT H T H TTT HHH TTT H TT H T H T HHH T HH TTT H TT HHHH TT
T HHHH T HHH TTT HH TT H TT HH T H T H T HH TT H T HH T H T H T H TTT H T H T HH T HHHHH T HHHHHH T HHH TT HHHHH T H T HH T H TTT HH TT HH T H TT HHHH TTT
HH T HH T HH T H TT HH T HH TTTT HH T H TTT H T H TT HH T HHHH TTTT H T H TTT HHH TT HHHH TT H T H TT H T HH TT HH T HH TTT HH TT HHHH T HHH TTT HH TTT
HH T H T H T H TTT HH TT HH T H TTT HHHHHHHHHH TTT H TTT HHHHHH TT H T HHH TTTTTT H TT HH TTTTT H T H T H T HH TTT H T HHH TT H TT H TTTT H TT HHH
TTTT HHH TTTTTT HH T HHHH TT HH T H T H T H TTT HH T HH T HH TT H TTT HHH TT H TTT HHH TTT HHH T H T H TTT H TT H TTTTT HH T HH T H T H T H T HHHH TT H
TT H T H T HHHH TTT H TT H TTTTTT HHH T H T H T HH T H TTTTTT H TT HHH T H T H T H T H TTTTT H T HH TTT H TTTT H T H TT HH TT H T H TTTT H TTTT H T H TT HH

 

If you asked most people if this looked random, they'd probably say no, because they see lots of long runs of the same face: two runs of Tails near the beginning, two runs of Heads in the middle of row 8. But this is actually the expected nature of random sequences. If there weren't many sequences like that, it would have to have memory so that it could avoid generating the same results that have been produced recently. If each event is independent there is no memory, so nothing prevents clustering like this.

 

This incorrect intuition is the source of the"gambler's fallacy": if it's been a while since you threw Heads, then Heads are "due" so you should bet on it.

[tx10s]

To make my point about clustering more concrete, suppose you threw a coin 1000 times, and the result is like this:

 

TTTTTTT H T H TTTTT HH T H TTT H T HH T H T H TT HH T HHHH TTTT H TT HH TTTT H T H TT H TTTTT HHHH TT HHHHH T HHHH TT HHH TTTT H T H TT H TT H T HH
H TT H T H T HH TT H T H TTTT H T H TTTT H T HH TT HHH T H TT HH T HH T HHHH TT HH T H TTTT H T H T H TT H TTT H TTT HHHHHH T H TTTTT HHH T H T H T HH T H TT
HHHHHHH TTT HHHH T HHH TTTT HHH T HH T HHH T HH TT HH T H TTT H TTT H T HH TT H T H T H TT H T H T HHHH T H T H T H TT HH TT H T HHHHH T H T H TTT H TTT H
H TTTTT HH TTTT H TT H T HHH TTT HHH T HHHHH TT HHH TT HH T H T HH T HHHH T HHH TTTTTTT H T HHH TTT HH T H TTTTTTT H TTT HHH TTTTTTTT HH T H
H T H T HHH TT H TTT HHHH TT H T HHHH T HHH T HH TTTTT H TTT H T H TTT H T HHH TTT H TTTTTTT H T H TTT HHH TTT H TT H T H T HHH T HH TTT H TT HHHH TT
T HHHH T HHH TTT HH TT H TT HH T H T H T HH TT H T HH T H T H T H TTT H T H T HH T HHHHH T HHHHHH T HHH TT HHHHH T H T HH T H TTT HH TT HH T H TT HHHH TTT
HH T HH T HH T H TT HH T HH TTTT HH T H TTT H T H TT HH T HHHH TTTT H T H TTT HHH TT HHHH TT H T H TT H T HH TT HH T HH TTT HH TT HHHH T HHH TTT HH TTT
HH T H T H T H TTT HH TT HH T H TTT HHHHHHHHHH TTT H TTT HHHHHH TT H T HHH TTTTTT H TT HH TTTTT H T H T H T HH TTT H T HHH TT H TT H TTTT H TT HHH
TTTT HHH TTTTTT HH T HHHH TT HH T H T H T H TTT HH T HH T HH TT H TTT HHH TT H TTT HHH TTT HHH T H T H TTT H TT H TTTTT HH T HH T H T H T H T HHHH TT H
TT H T H T HHHH TTT H TT H TTTTTT HHH T H T H T HH T H TTTTTT H TT HHH T H T H T H T H TTTTT H T HH TTT H TTTT H T H TT HH TT H T H TTTT H TTTT H T H TT HH

 

If you asked most people if this looked random, they'd probably say no, because they see lots of long runs of the same face: two runs of Tails near the beginning, two runs of Heads in the middle of row 8. But this is actually the expected nature of random sequences. If there weren't many sequences like that, it would have to have memory so that it could avoid generating the same results that have been produced recently. If each event is independent there is no memory, so nothing prevents clustering like this.

 

This incorrect intuition is the source of the"gambler's fallacy": if it's been a while since you threw Heads, then Heads are "due" so you should bet on it.

 

Well Mr. MIT (I graduated from Cal Tech myself), you apparently need a basic lesson in probability. As you know, with one finesse, (or coin flip), there are two possible outcomes, Onside (I will call Y, and Offside, I will call N). With two finesses, the combinations are NN, NY, YN, and YY, thus 25% will have no finesses, 50% will have 1 finesse and 25% will have 2 finesses. I am sure you understand that this can be expanded to any number of finesses, and a basic Excel spreadsheet can calculate the number of combinations for an number of finesses. I found that 20 tournaments had an average of about 280 finesses. 280 finesses have a total of 1.94267E+84 possible combinations for the 281 total outcomes (0 to 280 Y) These average 280 finesses had an average of 112 finesses in favor of the player. There are 9.5056E+80 combinations for 112 or less finesses. Thus, the chance of getting 112 or fewer finesses on side out of 280 total finesses is about 1 in 2000. My data showed the same 1 in 2000+/- outcome (within a fairly narrow margin)all 9 separate times. For your example, the chances of getting 400 or fewer heads in 1000 coin flips is about 1 billion to one. In other words, the greater the number of coin flips or finesses, the less likely the final count will be very far outside (as a percentage of the total) from the middle outcome, as in a bell shaped outcome distribution. When I did the analysis, I was not sure what the outcome would be, I just had a feeling that the hand distributions were not random, and unlike others, I actually accumulated the data to see what the distributions actually were. Your problem is you are looking at small cluster distributions rather than total outcomes, which is what I did.

If you would like to tell me what is wrong with my calculations, I am willing to listen. If you think I am lying or did not obtain the data correctly, I posted two sets of 20 tournaments with my calculated outcomes. I do not know if you are associated with BBO, but someone from there can double check my calculations if they want to.

BTW, I am not assuming some conspiracy that the finesse distribution just happened to change after my post. I had complained to BBO support service about the distributions and they said that their programmers would look at the issue, so I am assuming someone looked and "corrected" the situation. Since BBO has long insisted that the hands were totally random, I would not expect them to ever admit that they made the change.