probability distribution of high-card points

[judyo...@gmail.com]

For several years, I've been puzzled by this:

You have 5-3-3-2 distribution with 12 hcp.

What is the distribution of the long suit's hcp?

The computations I have tried are certainly wrong. As the hand's total hcp rises, my computation of the long suit hcp seems chaotic.

I *suspect* that having 0 out of 12 in the long sun is very rare. But I can't prove it.

Does anyone know a reference? (I am not very interested a simulation.)

Carl

[judyo...@gmail.com]

On Tuesday, September 28, 2021 at 3:08:52 PM UTC-4, judyo...@verizon.net wrote:

> I *suspect* that having 0 out of 12 in the long sun is very rare. But I can't prove it.

I have less confidence in this.

I've just done the computation for a 1-hcp hand shaped 5-3-3-2.

The probability that the jack is in the 5-card suit is 28 / 59 ~ .47.

Carl

[Don Reble]

> You have 5-3-3-2 distribution with 12 hcp.
> What is the distribution of the long suit's hcp?

> I am not very interested a simulation.

Perhaps an actual count, then?

There are 651552408 hands with 12HCP, 5 clubs, 3H, 3D, 2S.

- 22278942
0 22278942 0.034194

J 34145496
1 34145496 0.052406

Q 45532746
2 45532746 0.069883

QJ 38758356
K 58137534
3 96895890 0.148715

KJ 48945708
A 73418562
4 122364270 0.187804

KQ 57861972
AJ 57861972
5 115723944 0.177613

KQJ 24666768
AQ 57555792
6 82222560 0.126195

AQJ 24854688
AK 57994272
7 82848960 0.127156

AKJ 26062560
8 26062560 0.040001

AKQ 20036160
9 20036160 0.030751

AKQJ 3440880
10 3440880 0.005281

--
Don Reble d...@nk.ca

[Fabulous Pedigree]

I hope Don Reble won't feel insulted if I tell him what he already knows:
His numbers are correct!
This seemed like a fun little program to add to my fun source distribution;
it was very convenient to have Don's numbers to double-check my code.

Here's a possibly useful result for No Trump bidding:
With 16 HCP and 2-3-3-5, how many HCPs can you expect in your doubleton?
0 - 61235676 22.4%
1 - 23299596 8.5%
2 - 34276284 12.5%
3 - 51778800 18.9%
4 - 66772620 24.4%
5 - 16587972 6.1%
6 - 9555444 3.5%
7 - 10406196 3.8%

Email me at the From: address if you want any of this source.

Cheers,
Jamie

[judyo...@gmail.com]

I was interested in distribution of hcp in *long* suit.

Carl

[judyo...@gmail.com]

thank you.