Deal Master Pro

[Cascade]

Probabilities of the longest and shortest suits in a four-hand deal:

 

Longest Suit
4 0.0292944677905751
5 0.402336782879317
6 0.422075366415914
7 0.126442683507377
8 0.0183051683243252
9 0.00147821697795984
10 6.58450191573011E-5
11 1.45628565027458E-6
12 1.27745286982186E-8
13 2.51963123452264E-11

Shortest Suit
0 0.183766327186213
1 0.609953117477308
2 0.205349135658076
3 0.000931419678402451

 

These numbers were computed by considering every possible four hand distribution and calculating the probability of those four-hand distributions. The longest and shortest suits were noted and the probabilities summed accordingly.

[Cascade]

Binomial Distribution for the number of four-hand deals with an 8-card suit and an 8-card or longer suit in a session of 36 deals.

	
8-card	        8-card or longer
0.018305168	0.019850699
0	0.514226158	0.485870394
1	0.345186568	0.354247273
2	0.112639097	0.125553303
3	0.023803677	0.028818368
4	0.003661805	0.004815118
5	0.000436991	0.000624123
6	4.20998E-05	6.53077E-05
7	3.36434E-06	5.66854E-06
8	2.27408E-07	4.16162E-07
9	1.31922E-08	2.62218E-08
10	6.6417E-10	1.43387E-09
11	2.92723E-11	6.86394E-11
12	1.13714E-12	2.89612E-12
13	3.91452E-14	1.08285E-13
14	1.19916E-15	3.60289E-15
15	3.27947E-17	1.0702E-16
16	8.02603E-19	2.84478E-18
17	1.76067E-20	6.77818E-20
18	3.46543E-22	1.44903E-21
19	6.12172E-24	2.78023E-23
20	9.70263E-26	4.78611E-25
21	1.37844E-27	7.38528E-27
22	1.75248E-29	1.01981E-28
23	1.98907E-31	1.2572E-30
24	2.009E-33	1.37917E-32
25	1.79812E-35	1.34074E-34
26	1.41852E-37	1.14881E-36
27	9.79644E-40	8.6172E-39
28	5.87152E-42	5.60963E-41
29	3.02023E-44	3.13408E-43
30	1.31405E-46	1.48105E-45
31	4.74242E-49	5.80555E-48
32	1.38171E-51	1.83716E-50
33	3.12292E-54	4.51E-53
34	5.13807E-57	8.0594E-56
35	5.47469E-60	9.32713E-59
36	2.83566E-63	5.24722E-62

[Cascade]

So the probability in 36 deals of have three or more deals with at least one eight-card or longer suits is:

 

1 - 0.485870394 - 0.354247273 - 0.125553303 = 0.034329029

 

Edit:

 

The probability in 36 deals of having three or more deals with at least one eight-card suit (precisely eight) is:

 

1 - 0.514226158 - 0.345186568 - 0.112639097 = 0.027948178

[semeai]

Awesome data, Cascade!

 

Cascade's exact answer of 1.985...% is pleasantly close to hrothgar's simulation answer of 1.978...%.

[PrecisionL]

Awesome data, Cascade!

 

Cascade's exact answer of 1.985...% is pleasantly close to hrothgar's simulation answer of 1.978...%.

 

From bridge odds complete, by Frost, Kibler, Telfer, and Traub, 2nd edition, 1971:

 

f(x) 8-cards (exactly) = 0.117% (Specific suit, one player); 0.47% (One player, any suit); 1.87% (The Deal, any suit)

 

And, 8+ cards = 1.87 + 0.148 + 0.0066 + 0.00015 = 2.025 % (8 or more cards in a suit).

[wyman]

Great.

 

Cascade's answer is for 8 cards or longer (which is really what we should use here, since had you seen an 8, a 10, and an 8, you'd also be surprised, but I'm happy to use PrecisionL's low estimate).

 

So let p be the probability you want to use (whether it be 0.0187 or 0.01985 or whatever). Then the odds of seeing 3 or more deals is 1 - the probability of seeing exactly 0 - the probability of seeing exactly 1 - the probability of seeing exactly 2

 

Prob of seeing exactly N in 36 boards = 36CN * p^N * (1-p)^(36-N)

 

So your computation should look like this, when p=0.0187

http://www.wolframalpha.com/input/?i=1+-+Sum+of+[binomial[36%2Cn]+*+0.0187^n+*+%281-0.0187%29^%2836-n%29]+for+n+from+0+to+2

 

You get 0.0295, which is about 3%, so once in 33 sessions (or, as others have said, ~once a month for a daily game). If you were to include 9+ card suits, you get 3.4%, as Cascade mentions above.

 

So, this provides no compelling evidence that Dealmaster's RNG is broken. Also, keep in mind that as you look at the daily games to see what seems normal, you're running experiments. So you might say "oh my god, we got 2 days in a week where we got 8 card suits 3 times! What are the odds?" But if you've been watching the output for a year, you've actually run that experiment (look at one week, count days where we get 3 8-card suits) 52 times (and arguably more, since you can choose to start your week on Monday, or Thursday, or whenever you want). It's a dangerous game (see comic strip below).

 

http://xkcd.com/882/

 

Cheers.

[semeai]

And, 8+ cards = 1.87 + 0.148 + 0.0066 + 0.00015 = 2.025 % (8 or more cards in a suit).

 

This is off slightly; you're overcounting. One player could hold an eight card suit and one player could hold a nine card suit, for example.

[mrdct]

Sorry if it's been covered by the mathematicians already, but if one hand has an 8-card suit that must signicantly increases the chances that one of the other three hands also holds an 8+ card suit. Kind of like the Sith they follow the Rule of Two.

 

Notwithstanding that, the prevailing view seems to be that a set of 36-boards containing three 8-card suits has a probability of around 3.5% so it's quite reasonable to expect it to come up once every 30 sets or so. The OP has indicated that his sample is 60 sets so if this is the only occurence, it's actually happening less frequently than expected; but of course only 60 sets of boards is surely an inadequate sample size to draw any firm conclusions.

 

It's been a good 20 years since I last played bridge with hand-dealt cards so I can't really remember if hands were flatter back in the days of shuffling and dealing, but my local club has been using Deal Master Pro for years and I've never heard anyone complaining that the hands seem more distributional than expected.

[Cascade]

From bridge odds complete, by Frost, Kibler, Telfer, and Traub, 2nd edition, 1971:

 

f(x) 8-cards (exactly) = 0.117% (Specific suit, one player); 0.47% (One player, any suit); 1.87% (The Deal, any suit)

 

 

The bolded (my editing) number is slightly different than what I calculated. I have double checked my calculation and think it is correct. Can anyone confirm?

[PrecisionL]

The bolded (my editing) number is slightly different than what I calculated. I have double checked my calculation and think it is correct. Can anyone confirm?

 

Your calculations for 9 - 13 card suits agree very well with my reference.

 

For 0 - 7 card suits I find poor agreement.

 

(Before electronic calculators I had calculated the HCP frequencies and found exact agreement with the authors.)

[Cascade]

Probabilities of the longest and shortest suits in a four-hand deal:

 

Longest Suit
4 0.0292944677905751
5 0.402336782879317
6 0.422075366415914
7 0.126442683507377
8 0.0183051683243252
9 0.00147821697795984
10 6.58450191573011E-5
11 1.45628565027458E-6
12 1.27745286982186E-8
13 2.51963123452264E-11

...

 

These numbers were computed by considering every possible four hand distribution and calculating the probability of those four-hand distributions. The longest and shortest suits were noted and the probabilities summed accordingly.

 

 

Your calculations for 9 - 13 card suits agree very well with my reference.

 

For 0 - 7 card suits I find poor agreement.

 

(Before electronic calculators I had calculated the HCP frequencies and found exact agreement with the authors.)

 

I did a 100000000 deal simulation and the numbers agreed pretty well with my calculation.

 

    4    2930017
   5   40233065
   6   42210696
   7   12641140
   8    1830499
   9     147863
  10       6553
  11        166
  12          1
  13          0

 

Is it possible your source is calculating something different?

 

I am simply counting the frequency of the longest suit in the four hands.

[Cascade]

Sorry if it's been covered by the mathematicians already, but if one hand has an 8-card suit that must signicantly increases the chances that one of the other three hands also holds an 8+ card suit. Kind of like the Sith they follow the Rule of Two.

 

Notwithstanding that, the prevailing view seems to be that a set of 36-boards containing three 8-card suits has a probability of around 3.5% so it's quite reasonable to expect it to come up once every 30 sets or so. The OP has indicated that his sample is 60 sets so if this is the only occurence, it's actually happening less frequently than expected; but of course only 60 sets of boards is surely an inadequate sample size to draw any firm conclusions.

 

It's been a good 20 years since I last played bridge with hand-dealt cards so I can't really remember if hands were flatter back in the days of shuffling and dealing, but my local club has been using Deal Master Pro for years and I've never heard anyone complaining that the hands seem more distributional than expected.

 

Yes there is a conditional probability that given one eight card suit the liklihood of a second is higher than the a priori probability of an eight card suit. I haven't checked my numbers but I think given one eight card suit the probability rises to over 3% that another hand has an eight-card suit (from 1.8%). Perhaps this is not as much as you imagined Dave.

 

Its many years since I first analyzed the computer dealt hands from algorithms that I had written but my memory is that the most significant error was in not dealing enough distributional hands.

 

Even so the most common criticism I have ever heard of computer hands is that they are "too distributional" (or "wild" or some other synonym).

[PeterAlan]

Probabilities of the longest and shortest suits in a four-hand deal:

 

Longest Suit
4 0.0292944677905751
5 0.402336782879317
6 0.422075366415914
7 0.126442683507377
8 0.0183051683243252
9 0.00147821697795984
10 6.58450191573011E-5
11 1.45628565027458E-6
12 1.27745286982186E-8
13 2.51963123452264E-11

Shortest Suit
0 0.183766327186213
1 0.609953117477308
2 0.205349135658076
3 0.000931419678402451

 

These numbers were computed by considering every possible four hand distribution and calculating the probability of those four-hand distributions. The longest and shortest suits were noted and the probabilities summed accordingly.

The bolded (my editing) number is slightly different than what I calculated. I have double checked my calculation and think it is correct. Can anyone confirm?

 

FWIW, I calculated the same basic statistics a while ago and I'll confirm the Longest Suit proportions that Cascade quoted (and in particular that 1.83% rather than 1.87% is the probability that the longest suit in a deal is of exactly 8-card length).

 

For those that are interested in the full, gory details, of the total of 53,644,737,765,488,792,839,237,440,000 possible deals, there are:

 1,571,494,042,604,960,223,750,000,000 where the longest suit is  4 cards (in any of the 4 hands)

21,583,251,210,971,361,009,130,800,768 where the longest suit is  5 cards

22,642,122,348,654,241,172,787,919,872 where the longest suit is  6 cards

 6,782,984,599,117,957,218,132,857,856 where the longest suit is  7 cards

   981,975,954,511,555,218,232,092,504 where the longest suit is  8 cards

    79,298,562,143,148,725,113,050,600 where the longest suit is  9 cards

     3,532,238,785,856,985,879,197,952 where the longest suit is 10 cards

        78,122,061,820,624,147,552,512 where the longest suit is 11 cards

           685,286,242,093,646,948,664 where the longest suit is 12 cards

             1,351,649,568,417,019,272 where the longest suit is 13 cards

These lead, as he says, to his figure of 3.43% for the probability of 3 (or more) deals in a sample of 36 deals containing at least one suit of 8 (or more) cards (and to a probability of 2.88% of exactly 3 such deals: so there is a non-negligible probability of 0.55% of there being 4 or more such deals out of the 36).

 

To give this number (3.43%) some perspective, it's rather larger than the probability (2.99%) of picking up a 4-4-4-1 hand in any one deal. I don't suppose most of the club members would believe that could be true: it just shows how our intuitions (preconceptions? prejudices?) can lead us astray.

[stwafl]

You know, everyone has complained for years about the distributions found in computer dealt hands. Most people ask the question 'are they right'. However, why not ask the question 'why does everyone think they are wrong'? A paper by the Canadian Journal of Statistics titled 'On the Distribution of Hand Patterns in Bridge: Man-Dealt versus Computer-Dealt' does just that - and explains WHY we all think the computer hands are wrong when they are not. The paper can be found in the JSTOR archives (www.jstor.org). Well worth reading.

[bluecalm]

What a fine example of circular reasoning: You use a random deal generator to check whether a random deal generator delivers the correct amount of 8 card suits. It is like saying that snow is white (or red) because it has the same color as snow which we know is white (or red).

 

Lol, you are completely clueless.

Some random number generators are proven to be good.

We don't have that informatino about dmpro so we use proven random number generator to check if dmpro give results with acceptable margin of error.

 

And most likely, your random generator in Matlab is not better than the random generator Deal Master Pro uses.

 

Dealing random bridge hands is trivial task assuming we have good random number generator (which we just use to shuffle number from 1 to 52 and assign first 13 to 1st hand etc.).

Now, dealing random bridge hands fast might not be that trivial task. It's quite likely that dmpro author used some not perfect algorithm for dealing hands in order to deal them fast enough. What other poster tried to show you is that what dmpro generates corresponds to what simple correct algorithm generates.

 

WHY we all think the computer hands are wrong when they are not

 

Imo you have to be quite naive to think computer generated hands are "wrong". Staking your intuition shaped be few thousands hands you might have played during your life against trillions easily generated by proven random numbers generators is sign of either ignorance, arrogance or most likely mix of both.

 

Now I am not saying you shouldn't question corectness of algorithm or admit your intuition doesn't agree with what computers generate but if you make a jump from that dissonance to opinion about correctness of generated deals you need to reexamine your way of shaping opinions.

[han]

This is an entertaining thread, but could you please all quote the name of the person you are trying to insult? That makes for far easier reading. I have no further complaints, please continue!

[mycroft]

If you have a good random number generator, don't use any "shuffle" algorithm. Generate 96 bits of random information and look it up in the Big Book, say the Andrews version, or the Pavlicek version. (Needless to say, the Big Book doesn't actually exist, it's an enumeration of hands function, but it's still better to use the 1:1 mapping than any "shuffle" function that isn't the 1:1 mapping).