Re: QT9xxxx A J KJxx hand again

paulhigh@xxxxxxxx wrote:

As far as this being a "small simulation", the difference between 200
hands and 1000 hands is that the confidence interval will be smaller
by a factor of the square root of 5 (the ratio of the sample sizes)
with the larger simulation. A 95% confidence interval for making game
in spades opposite would be (63/200) = 31.5%, plus or minus 1.96 times
the square root of (.315)(1-.315)/200, which is plus or minus
6.4% .You can be 95% confident an infinitely large simulation would
give a proportion between 25.1% and 37.9% . If a sample of 1000
yielded the same proportion making (31.5%), the confidence interval
would shrink to plus or minus 2.9%, or 28.6% to 34.4% . When the
decision is whether or not to bid a close game or slam, the difference
between plus or minus 6.4% and 2.9% may be significant. Not sure it
matters much here -- game is clearly a bad bet, slam is possible but
unlikely, opening 4S is aimed at making the opps guess rather than
scientifically bidding to our optimum contract.

You can use this formula for sample sizes of 200 or more: Let "p" be
the observed sample proportion of making contracts, and "n" be the
sample size. Then a 95% confidence interval is given by p plus or
minus 1.96 times the square root of p*(1-p)/n . For sample sizes below
200, the 1.96 wouldn't be correct, you'd need to look in a table of
Student's "t" distribution (it will be larger the smaller the sample
size.) At or above 200, the 1.96 is valid. (This is a measure of how
wide the confidence interval is in terms of standard deviations.) A
very simple, statistically accurate formula, which allows you to make
a definite statement rather than a vague disclaimer regarding sample
size.

Note carefully that I claimed an infinitely large simulation would
probably yield results in a certain range. I said nothing about table
results, as the relationship between simulations and table results is
still unclear.




In actual fact, the number required for a full sample is far from infinite, since the number of ways the four hands can be distributed is less than 5.4 time 10 to the 28th power and my calculator says the way that the other hands can be distributed once you know one hand looks to be about 8.3 times 10 to the 16th..... which means practically infinute to someone sitting at the table trying to figure things out at the moment.


However, it is good to know that the next time you hold this particular hand in this particular position you will have a better idea as to what to do.

Bob
.

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