Re: HCP distribution in a 2NT opener

Frances wrote:
> Reef Fish wrote:
> > Lorne wrote:
> > > "Reef Fish" <Large_Nassau_Grouper@xxxxxxxxx> wrote in message
> > > news:1125611325.077845.14640@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> > > >
> > > > Lorne wrote:
> > > >> "Sartaj Hans" <spadedeuce@xxxxxxxxx> wrote in message
> > > >> news:1125559372.435427.57190@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> > > >> >I recall reading (probably in this newsgroup or BBO )that a 2NT opener
> > > >> > 20-22 HCP will have 20 HCP 60 percent of the time.
> > > >> > That doesnt feel intuitive.
> > > >> > Can someone please clarify either way ?
> > > >>
> > > >> For what its worth my simulator came out with:
> > > >>
> > > >> 20: 52.6%
> > > >> 21: 29.8%
> > > >> 22: 17.6%
> > > >>
> > > >> run over 10,000 random deals which contained a 2N opener for one of the 4
> > > >> hands.
> > > >
> > > > I have to admit I am surprised that it came that close to the
> > > > percentages without restricting the NT hands to balanced
> > > > distributions.
> > > >
> > >
> > > It was hands that would open 2N so they will be 4432/4333/5332/5422. The
> > > program does not treat 6 card minors or 4441 as 2N openers.
> > >
> > >
> > > > Which were the 4 types of balanced distributions admitted?
> > > >
> > > > Was the sampling done by eliminating unsuitable candidates until
> > > > a total of 10,000 suitable random hands are kept? That would
> > > > have required a similation of approximately 800,000 random deals.
> > >
> > > My guess is that it did run at least that many, maybe closer to 850,000
> >
> > According to the known distribution of HCPs given by Rik and Pavlieck,
> > 1.23% of all deals result in 20-22 HCPs, and these INCLUDE the hands
> > you kept, so the expected number of similated hands to fill the 10,000
> > hands is 10000/.0123 or about 813,000.
> >
> > > >
> > > > Did you do it in some other (more efficient) ways?
> > > >
> > >
> > > Takes less than a minute to run It deals, checks the 4 opening bids, if
> > > nobody bids 2N it redeals until somebody does (average 80-90 deals to get
> > > each 2N bid)
> >
> > Actually because of the very different probabilities of getting each
> > type of hands by HCPs, it takes more than 3 times the number of deals
> > to get one 22 HCP hand than that of 20 HCP. The expected number of
> > HANDS of each type to make up your 10,000 total is, approximately,
> > 5232 (20 HCP), 3072 (21 HCP), and 1708 (22 HCP). These do not sum
> > to exactly 10,000 becaues of roundoff in my results.
> >
> > Is the program you used commercially available? I have not paid
> > much attention to various simulation programs. The late Paul Heitner
> > used to simulate almost everything for his comments for the answers
> > to the Master Solvers' Club bidding questions in the Bridge World.
> > For many difficult to intractible probability problems of this type,
> > the monte carlo method is certainly the say to go, to get the
> > numerical answers correct to any number of decimal places by
> > cheap labor of machine cranking.
>
>
> This particular example is not particularly difficult.
> I've given exact percentages above for my definition of a balanced
> hand, and am happy to re-do them for any other.

You are quite correct.

Lorne's definition of a balanced hand was:

L> It was hands that would open 2N so they will be 4432/4333/5332/5422

Using Table 1 in the Encyclopedia for the distribution of hand
patterns,
restricted to those four types, I was able to get the exact percentages
for those balanced types to be

52.258552 for 20 points, vs 52.6 from Lorne's simulation
30.684847 for 21 points, vs 29.8 from Lorne's simulation
17.056602 for 22 points, vs 17.6 from Lorne's simulation

But it took more than 1 minute :-) (even with the Table of
distributions
given in the Encyclopedia) to get those exact percentages, though not
as tedious as I had thought.

-- Bob.

>
> (Certainly many similar problems are extremely difficult and a
> simulation is the best approach.)
>
> >
> > One interesting fact about you sample estimates of the proportion
> > of 20, 21, and 22 HCP hands all have standard errors of .00111
> > <sqrt(pi*(1-pi)/ni)> from the way your hands are simulated and
> > tallied.
> >
> > -- Bob.

.

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