[KimH56]

Can someone confirm my findings on the probability of holding five clubs when opening a Std. Am. (better minor) 1C? tx Kim

[manudude03]

KimH56, on 2011-August-05, 17:28, said:

Can someone confirm my findings on the probability of holding five clubs when opening a Std. Am. (better minor) 1C? tx Kim

According to this site, its approximately 63%. Note that thats for 5 cards or longer. I can't give you the figure for exactly 5 clubs.

[jdeegan]

manudude03, on 2011-August-05, 18:00, said:

According to this site, its approximately 63%. Note that thats for 5 cards or longer. I can't give you the figure for exactly 5 clubs.

63% seems too high at first blush, but it is a GREAT question!!! Sometime soon, I will break out my Pavlicek random hand generator and see what I can do. There is no real definitive single answer since one has to make specific assumptions about one's system, but certainly a general conclustion can be drawn.
The opponents' bidding can also be a big tip off in a competitive auction. For example, it never bothered me that pard might have opened a three card diamond suit with 4-4-3-2 distribution since it rarely came up, AND if the opps compete in a major where I have length, the the probability of that hand almost vanishes.

[barmar]

I'm surprised that playing weak NT has so little affect on the probabilities of minor suit openings.

But now I realize I'm thinking of it the wrong way. NT range changes the probability that you'll open a minor vs 1NT. But given that you did open a minor, it doesn't affect the probabilities of the suit length too much.

[hotShot]

Since NT takes away balanced hands you'd expect more unbalanced hands, and since the restriction is to 1♣ opening, ♣ will be the longest suit.
The 63% are there because it is an additional restriction that you open 1D holding 4-4 in the minors.
This reduces the number of hands that could be opened in 1♣ in general and especially those that have 4♣.

[inquiry]

I doubt the answer is anywhere near 63%, The webpage that this number comes from doesn't show work, but still my experience doesn't seem to bear this number out. I open a club a lot, and my experience is that about half the time, maybe more, I open with only three or four clubs.

The largest single player database I have is pulled from JEC matches. I took a look at 1763 hands jec and his various parnters opened 1♣ (Mr. Cayne plays standard american).

302 of these hands they held 3 card suit
494 of these hands they held 4 card suit
612 of these hands they held 5 card suit
292 of these hands they held 6 card suit
60 of these hands they hald 7 card suit
3 of these hands they held 8 card suit

This is 45.1% opened with 3 or 4 clubs, 54.9% opened 1C with five or more clubs. This percentage with long clubs here is slightly higher than my "gut" feeling it should be, but still WAY lower than the quoted 63%.

I know how to use combinations to find how many possible hands fall into the requirement that the hand pattern be 4333 or four club-not four-diamonds-32, or clubs longest suit starting at five. With excel, these hand patterns are easy to calculate (once you get to seven clubs, ignore other suits, and just lump them all together).

You can try to approach this problem mathematically (others on the forum are much better at this sort of thing than me). You have to deal with combination in excel, this is the combin(#1,#2) thing. There are 286 ways you can draw out three clubs from the 13 in the deck (combine(13,3)=286, there are 715 ways to draw four cards in a suit.

So to find total hands numbers, you have, 3=3=4=3, 3=3=3=4 is 286*286*715*286 plus 286*286*286*715
Then with 4 clubs, not 4 diamonds, you have 4=3=3=3, 4=3=4=2, 4=3=2=4, 4=1=4=4, there are only 78 ways to take two cards out of 13
Then with 5 clubs, no other suit as long as five cards, you get a combination these patterns 5=3=3=2 (times three), 5=4=4=0 (times 3), and 5=4=3=1 (times six). Combination with 1 card out of 13 is one, comination for 0 cards is one (you have no card).

Once you get to six club, no other six card suit, the distributions become
6322 (times 3)
6421 (times 6)
6511 (times 3)
6502 (times 6)
At seven or more clubs, you no longer have to worry about another suit as long or longer than clubs. So the match is a little easier. It is a combinaiton for the number of clubs (for instance C(13,7) or C(13,8)), and then total number of remaining cards out of the 39 non clubs. So you can do this
C(13,7)*C(39,6), and C(13,8)*C(13,5). I think it is probably not worth looking at the 9 or more clubs (surely you wouldn't open 1C holding 12 clubs, for instance).

So,,,
3334 pattern times three (only 3 diamonds allowed)
4432 pattern times four (allowed only two or three diamonds)
4144 pattern only once
5332 patter times 3
5422 pattern time 3
5431 pattern times 6
5440 pattern times 3
6322 pattern times 3
6331 pattern times 3
6421 pattern times 6
6430 pattern times 6
6511 times 3
6520 pattern times 6
tnen just the combinations for 7, 8 and 9 clubs given above.

If I didn't make any mistakes, when I did this math, I end up with some some very large numbers. The total number of possible hands (just based upon card combinations with restrictions listed above) is 1.97 x 10^11, the number with 3 or 4 clubs is more than half that (1.005 x 10^11). So it is 50% of the possible hand combinations. Of course, no influence on the chance that you might open 1NT is included in these numbers. Obviously with some of the hand patterns including some of the ones with five clubs and even six clubs, you might open 1NT if the point count is adequate. I haven't quite figured out how to approach this open 1NT with balanced hands (including 5332 and 6322 as balanced) in the calculations.

Maybe if you never open 1NT with 5332 and 6322 patterns the number might get to 60% or so. Anyone know the right approach to factor in opening values into the hand pattern calculations? This I could try to approximate using a large Bridgebrowser database, where I looked at balanced hands from 10 to 20 hcp that were opened something. What was the frequency of those hands that had 15-17 hcp.

For this quick look, I looked included hands with at least 4333 and 4432 distribution in a database of hands (just 2000 hands each category to do this quick). 32% of the hands that opened with those distributions had 15-17 hcp. Then I looked at 5332 hands, 31% of those fell into the 15-17 range, then I looked at 6322, which was also (rounded up) to 31%.

So of the none 3/4 hands (not counting 4144), 32% would open 1NT,
of the 5332 hands and 6332 hands 31% would open 1NT. If this stats hold, you would find.

These corrections lead to a calculated 55.1% of the 1♣ opening bids having 5 cards or more (based on questimates from a database for frequency of balanced hands with 15-17 hcp. I don't know if this is shear luck or something significant that this number is almost exaclty what the JEC bridgebrowser database showed when looking at JEC (and his parnters) opening 1♣ held in club suit (54.9% for him).

[hotShot]

How do you open a hand with 4♦ and 5♣ in Standard American?
If you open it 1♦ than the frequency of a 5 card ♣ opening is reduced.
The quoted website did not list this as restriction so I guess they did not use it.

[hotShot]

Just ran a few simulations and I can't get anywhere near the number from that website.

[kenberg]

Ben's numbers from the Cayne matches interest me. I don't know how much work that entailed but if it wasn't too great, maybe you could do the same for the 1D openings?

Looking at the site's numbers I was, at least initially, surprised by the closeness of the figures for a four card suit when the opening is 1♣ or 1♦. If a hand contains only one four card minor, that minor is, of course, as likely to be clubs as diamonds. But all of the hands where the minors are 4-4 are opened 1♦, and I would have expected this to have more influence than it does.

Comparing diamonds with clubs after a 1m opening, the site's given probability of a three card holding is much smaller with diamonds than with clubs, as would be expected if the only three card diamond opening is on 4-4 in the majors. The difference in probabilities for three card holdings has to be reflected in differences elsewhere, but it seems to be almost entirely in the five card holdings.

This seems a little odd.

[manudude03]

Hmmm I just made a spreadsheet with Ben's 32% figure for NT openings (31% for 5332) and I'm getting a figure of just over 50%.

My Spreadsheet

Note I didn't include 6322 for opening 1NT, and 4♦5♣ hands were opened 1♣ to give it the most favourable circumstances. If anyone sees a mistake in my workings, let me know.

I doubt the figures are 100% accurate. Balanced hands tend to have more HCP than unbalanced hands (as an extreme example, you can't have 17 HCP and 12 clubs). It's difficult to determine the unbalanced HCP range anyway (just look at all the "Open 1x or 2C" threads or "how many x" threads)

[helene_t]

Simulations of 8323 hands:
3: 19.7%
4: 31.3%
5: 27.6%
6: 17.0%
7: 3.7%
8: 0.6%
9: 0.04%
10+: never in this sample

The definition was:

open1c = function(hand) { return(suitlength(hand,clubs)>=3 && suitlength(hand,diamonds)<=3 && suitlength(hand,hearts)<=4 && suitlength(hand,spades)<=4 && hcps(hand)<=21 && hcps(hand)>=11 && (!balanced(hand) | hcps(hand) %in% c(12:14,18:19)))}

so this is somewhat oversimplified but it shouldn't be too far off.

[hotShot]

My simulation

Quote

Out of 1,000,000 deals 91244 fit the restrictions:
HCP 12-22, ♠0-4, ♥ 0-4, ♣ length > ♦ length or (♣ length = ♦ length and ♦ length not 4)
excluding strong NT defined as:
HCP 15-17, ♠ 2-4, ♥ 2-4, ♦ 2-5 and ♣ 2-5

Count    CL len.	%
17111	3	18.7530
26525	4	29.0704
30531	5	33.4608
13697	6	15.0114
2929	7	3.2101
417	8	0.4570
29	9	0.0318
5	10	0.0055

♣ length 5+ => 52.14%

[hotShot]

@ kenberg my simulation for ♦

Quote

Out of 1,000,000 deals 96586 fit the restrictions:
HCP 12-22, ♠0-4, ♥ 0-4, ♦ length > ♣ length or (♣ length = ♦ length and ♦ = 4)
excluding strong NT defined as:
HCP 15-17, ♠ 2-4, ♥ 2-4, ♦ 2-5 and ♣ 2-5

count	DI len.	%
4405	3	4.5607
40545	4	41.9781
34878	5	36.1108
13471	6	13.9472
2889	7	2.9911
366	8	0.3789
31	9	0.0321
1	10	0.0010

♦ length 5+ => 53,4%

[helene_t]

hotshot gets a lot more 5-card suits than I do. Hotshot opens 1♣ with 20-22 balanced and also with 5-5 minors and with 4♦5+ clubs, which I do not. In SA you open 1♣ with 4♦6♣, possibly with 4♦5♣ and 5bananas6♣ but not with 5♦5♣.

Changing my definitions to open 1♣ with 4♦5+♣ gives:
3 4 5 6 7 8 9
17.6 27.8 32.4 18.0 3.6 0.6 0.0

[hotShot]

helene_t, on 2011-August-07, 07:02, said:

hotshot gets a lot more 5-card suits than I do. Hotshot opens 1♣ with 20-22 balanced and also with 5-5 minors and with 4♦5+ clubs, which I do not. In SA you open 1♣ with 4♦6♣, possibly with 4♦5♣ and 5bananas6♣ but not with 5♦5♣.

Changing my definitions to open 1♣ with 4♦5+♣ gives:

 3   4    5 	6   7   8   9
17.6 27.8 32.4 18.0 3.6 0.6 0.0

Even with that we are about 10 %-points away from that website result, so I guess that it is wrong.

[kenberg]

I started to say that Manadude allows 1♣ openings on five clubs and four diamonds while Helene does not but fortunately I checked before posting and found Helene already responded.


Clearly the results vary, fairly significantly, with the assumptions made. I would criticize the site http://www.dur.ac.uk...statistics.html on two grounds: Giving the probabilities to five sig digs is misleading, and the assumptions behind the calculations are not completely stated. You can sort of infer that with five clubs and four diamonds they assumed 1♣ but it is not certain.

Data can mislead, and Ben's 1763 hands have to be treated cautiously, but still I would expect that a simulation that reflects the jec approach should come reasonably close to Ben's figures. Or to put it another way: Are the discrepancies due to insufficiently many hands or due to the simulations not capturing the jec approach?

I need to think more, but for the moment I like the Manadude figures.

[inquiry]

I took a look at this question not because I felt it was an interesting question, but because 63% seemed way to high to me, and I couldn't figure out how the website quoted got that number. For my calculation I threw in hands with 4 diamonds and 5 clubs as a 1♣ opener, although I tend to open these 1♦.

i think the simulations, the "calculations" and the real world experience seems to support the view that 63% is much too high, that is good enough to satisfy me. I am comfortable with an number in the mid 50% (53, 54, 55, 56, I would not argue against any of those).

As requested above, I took a look at the JEC match 1♦ openings. A few things surprised me. The first, is they opened 1♦ more often (well probably not significanlty more often) than 1♣. I never expected that! Here are the numbers.

94 - three card diamond suit
737 - four card diamond suit
618 - five card diamond suit
250 - six card diamond suit
69 - seven card diamond suit
5 - eight card diamond suit

So for this dataset, the 1♦ opening bid had a lower percentage of hands with five card suit (53% versus the 54.9% for 1♣). For these hands, there were six hands that held four diamonds and five clubs. On these six hands, they opened 1♦ all six times. The calculations I used had those hands opening 1♣ (see my post above). They had a total of 1,773 1♦ opening bids to 1763 1♣ openings. If we move the six 4=5 hands to 1♣ openings there would be near balance between the two. The thing is, they should not even be close to balanced. Why? Because of the 205 hands they held 3=3 in the minors (and 4-3 either way in majors) that they opened a minor, they opened 1♣ all 205 times. At first, I was unsure as to why the 1♦ opening should be so frewquent. And thought something might be wrong with the BBO dealing program. Now I am convinced this is to be expected. More below.

If we move the six 4=5 hands to the 1♣ opening pool from the earlier post that would raise the frequency of five card suits from 54.9% to 55%, essentially equaling the "calculasted value". Another thing I noticed is that they upgraded three hands with 5332 or 6322 and 14 hcp 1NT. If we move those from 1NT to 1♣ (for comparision to calculated value). However, they also downgraded 22 hands with 3=3=3=4. and 4-3-3=3 distribution and 18 hcp that would have opened 1♣ in the calculated group, which would decrease the frequency of short majors a bit (they only opened 1 4333 hand pattern with 14 hcp 1NT (3rd position, non-vul verssus vul, by a very good player, sillafu. The hand was ♠AKJ ♥KT75 ♦972 ♣KT2, you decide upgrade, or strategic decision).

So back to math. Is the BBO dealer flawed, or was I overlooking something. For 1♦ opening, they would not open 1♦ with 3=3 in the minors, so... To keep this simple with a five card or longer major we open the major unless there is a two card difference in the total.

3-4-3-3 times 1 pattern (just 4 diamonds)
3442 times 6 patterns (this is an increase. when clubs, you could not hold 4 diamonds, but with 4 diamonds, you can hold any other 4 card suit)
4144 only once (singleton club)
5332 six three patterns (same as for club opening)
5422 one extra pattern. You can have 5 diamonds and any of the others suits, plus four diamonds and five clubs (the extra one)
5431 same as for clubs, plus two extra patterns for 5club, four diamonds and 3-1 either way in majors.
5404 pattern is times 5 (2 EXTRA patterns for when holding five clubs and 4 diamonds plus three patterns for when holding five diamonds)
5521 pattern times two (five five in minors) all are additional patterns when compared to clubs
5503 pattern times two (five five in minors) all more additional hand patterns
6511 increase as 65 either way rather than just six clubs
6520 increase as 65 either way in minor, rather than just clubs.
6610 pattern twice (more hand patterns, 6-6 minors)
7, 8, 9 unchanged. except we agree to open 1M with 7 diamonds and 6 major.

The increases for the frequencies of 4-4 in the both minors and 55, 54, 65 in the minors more than make up for the loss of 33 in minors being opened 1C.

Basically, due to ties go to diamonds, or even 5C-4D or 6C-5D goes to diamonds, there are more hand distributions that can open 1D than 1C. However, since 4-3-3-3 and 3-4-3-3 are two of the most common hand patterns, this increase in patterns is overcome somewhat by the frequency of those hands. Making the opening about even. I have to admit this finding really surprized me. All of this results in an increase in the number of possible hand patterns that would open 1♦. The total number of combination are 2.025 x 10^11. The number quoted earlier for 1♣ opening include 5C-4D and 6C-5D and 7C-6D which are now included in the 1D openings. So if we remove those distribution, the 1.97 x !0^11 is decreased to 1.77 x 10^11.

So the ratio of 1C to 1D openings might be expected to be 1.77/2.025 or 1 to 1.14, or open 1D slighly more often than 1C in standard american context. That surprises me, because I am use to seeing a lot more 1C opening bids in open databases. But a lot of people play strong club, or open 1C with all balanced 11-14, etc (even when holding 2C) which throws that off I guess. Something is probably off with my math or assumptons about what would be opened, or I might have removed too many hand patterns from the 1C opening pattern, because the 1C/1D opening by JEC and partner was much closer to 1 to 1. But it was interesting to look.

[manudude03]

A couple of minor mistakes (I think anyway, our total hands seem way off for 1C seemed way off*) in there that I see:

You haven't mentioned for 4432 (3 diamonds) for the 1D opening.
4144- we would open 1D unless the singleton was diamonds no?
5332- you're double counting once you force the 5 card suit to be diamonds.
5404- Do people really open 1D with 4045 when you can easily rebid 1S over 1H?

*I just checked the total with the NT hands included and I get 1.98 x 10^11 with the hands with 4D5C, 5D6C etc included.

[inquiry]

manudude03, on 2011-August-07, 13:28, said:

*I just checked the total with the NT hands included and I get 1.98 x 10^11 with the hands with 4D5C, 5D6C etc included.

I had 1.97 x 10^11 when I included 4D5C and 5D6C in with the 1C openers. Not sure it that is rounded or not. 5332 hands there are six of them
2S3H5D3C, 3S2H5D3C, 3S3H5D2C. I included all of those.

I am very curious at how the bridge statistics web page quoted earlier came up with several of their numbers. For instance, that page says a 1♦ opening bid will have 5 or more diamonds 72% of the time rather or not you play weak or strong 1NT. I have tried to figure out how he came up with is numbers. It looks like he is taking the percentage of different hand patterns and multiplying by the percentage of hcp, but I have not been able to do this criss-cross and get his numbers. Anyone see how he derived these very high numbers?

[hotShot]

Just checked the 1♣/1♦ ration of my simulation if I move the 4+♦/ ♦len.+1♣ hands to the 1♦ opening.

I get the ration of 1: 1.13 which is close to inquiry's expectation of 1 : 1.14.