Newsgroup sci.stat.math 12347

Directory

"Paul R. Garvey"  writes:
>--
>Any opinions out there on the fuzzy methods and their relationship to
>probability theory?
There are several discussion papers out there.  Two series were:
Laviolette, Seaman, Barrett, and Woodall (1995) in Technometrics (and
ensuing reactions)
Special issue of IEEE transactions on Fuzzy Systems (feb. 1994)
(Fuzzines vs. probability)
But, I see you asked for "opinions" not citations.
I am no expert, and I understand why the probabilists eschew fuzzy set
theory, but I also know that fuzzy set methods are for *practical*
situations where preciseness is not only not required, it isn't desired.
For example, a dishwasher only needs to record temperatures of hot, medium
hot,...,cold...and it is effient for its purpose.  Any more precision is
expensive. Although, probabilists might say we can put a probability 
distribution on the imprecision...and everybody already accepts the laws 
of probability...not so for fuzzy membership
In the above papers, it is my opinion that the probablists "won"...by a
large margin (and I'm not just saying that because one of them is my
professor).  But, something tells me the two are fighting different
battles...as alluded to by some of the discussants.

bm373592@muenchen.org (Uenal Mutlu) wrote:
>I think this posting contains useful hints for a successful play 
>strategy! Read on!
>
>On Thu, 9 Jan 1997 14:38:13 GMT, nveilleu@NRCan.gc.ca (Normand Veilleux)
>wrote:
>
>>consecutive drawings if you buy 1 ticket per draw.  And even after 168
>>draws, there still is 0.042398 probability of having lost all draws.
>
>That's saying 95.76 % chance of winning (>= 3) if playing the same 1 ticket
>in 168 consecutive draws. IMHO an important conclusion from this would be:
> Playing the same 1 ticket in x consecutive draws is better than playing 
> x different tickets (or a wheel) in 1 draw. 
>Isn't it?
No. If the odds of winning a game are 1 in 54 million, then if you 
buy 54 tickets, your odds of winning are 1 in 1 million. If you buy
1 ticket for 54 drawings, your odds of winning never get above 1 in
54 million. The only way to increase your odds of winning is to buy
more tickets. The only thing that is important about them is that 
none of them have all of the same numbers. (Although, if the pot were
split, you would get 2 shares; not really worth considering as a 
strategy though.)
>If yes, then the further practical generalization of this statement 
>would be:
> Don't change your numbers; ie. play always the same numbers (tickets or
> wheel) until you have a win.
It doesn' matter. However, people who play the same numbers every 
week do tend to be more loyal players; they would probably want 
to kill themselves if their numbers came in, and they didn't play 
them that particular week!
> --> So one should also very well think of analysing the past draws for 
>     choosing the 'right' expected numbers (it's normally a one-time task)
>
>I think, that's it! Ie. IMO this is a very important key fact for a
>successfull play strategy! Isn't it?
There are no expected numbers, unless you mean the ones you play 
every week and are hoping they will be drawn.
>Sure! Because, we no longer start again from the beginning at each draw.
>Instead we keep it constant since probability says "using the same 
>numbers a win should occur in the next x draws..." But, if we change the
>numbers each time then everything starts again from the beginning, so this
>should be strictly avoided!...
>
>What do others think on this strategy?
If you want to increase the odds of winning, buy more tickets. 
The game is constructed so that those who play more are more 
likely to win. That is the incentive to play. And if you don't
play, the odds of winning are 0. If you bought all 54 million
combinations the odd of winning would be 1 in 1. (If you share
the pot you will win less than the total jackpot, so there is
no guarantee that you will get your 54 million back.) :-)
>>If you do come up with the same number, then it implies that wheeling
>>does not change, in any way, the average number of winning tickets.
>
>But then also the opposite is true: wheeling is at least equal to using
>the same number of any different randomly chosen single tickets. True?
As long as they are for the same drawing. Otherwise, wheeling
works better. But at the same time, saying "give me 168 easy
picks" is just a good. (You may not get a free ticket, but
if that is the only return on 168 bucks, it is not much of a 
consolation prize. That, and there is the fact that in 168 easy 
picks, two may be the same.)
>Are there any situations where wheeling behaves worser than using 
>randomly or even any some otherwise chosen different tickets of 
>same size?
If you think you have some psychic abilities, letting numbers 
pop into your head may work better then these "covering 
combinations" schemes. What is so sad is that even if you use
one of these methods of picking numbers and win the jackpot,
all it proves is that you "got lucky".
-- 
Craig
clfranck@worldnet.att.net
Manchester, NH
The less a man makes declarative statements, the
less he's apt to look foolish in retrospect.
  -- Quentin Tarantino in "Four Rooms"

On Fri, 10 Jan 1997 09:22:30 -0700, Karl Schultz  wrote:
>C. K. Lester wrote:
>> 
>> In article <32D53060.B3D@fc.hp.com>, Karl Schultz  wrote:
>> >C. K. Lester wrote:
>> >>
>> >> In response to Karl Schultz's prior post,
>> >>
>> >> >There are no subsets.  The 168-ticket wheel will guarantee a 3-match
>> >> >in a 6/49 lotto.
>> >> >
...
>The perceived value, IMHO, is as follows.  People like to win.
>If they can be sure to walk away with something, then they
>might take steps to do that.  The only way to increase your
>chances of winning is to play more numbers.  If you are
>in the habit of playing 100+ numbers at a time and have
>had a long losing streak, you might be inclined to play
>the 168-ticket wheel, so that you are sure to have to make
>that trip to the counter to claim a prize.  Actually, you
>have a 60+% chance of getting 3 wins with 168 tickets,
>but that is another story.  So, it is a psychological
>thing - sure to get a win.
I get a different percent value:
  Since the probability for at least 3 is about 1 in 54 (to be exact
  p=0.0186375450020), using the usual formula gives:
    E = "at least once at least 3 matching nbrs in 168 consecutive 
         draws using the same fixed 1 ticket" 
    p(E) = 1 - (1 - 0.018637545)^168 = 0.9576
    meaning 95.76 % of chance of occuring of the event E.
(BUT: I'm still not sure if this is the same as playing 168 different 
tickets in 1 drawing) 
Which formula did you use?
>In the end, you are right.  Big Deal.
>The wheel is just a structured way to buy more
>tickets, which, in itself will increase chances.
>Now, here is a real tough question for wheel experts.
>
>If one plays 168 tickets using the wheel, they are sure
>to match 3 at least once.  What does this wheel do to
>one's chances to match more than 3???  
One would need to analyse its guarantee table; I think Adolf 
mentioned of doing this in another posting:
:From: Adolf.Muehl@univie.ac.at (Adolf Muehl)
:Newsgroups: rec.gambling.lottery
:Subject: Winning chances of the 168 wheel from Uenal
:Date: 10 Jan 1997 13:18:29 GMT
:
:There was some discussion recently on the winning chances of the
:168 ticket 3/6 cover from Uenal.
:I have set up a program to calculate these winning chances. Unfortunately 
:I do not know an other way than to compare each of the
:13 million + possible tickets to the 168 tickets of the wheel(-:
:So there is much to compare, hopefully over the weekend results will be
:available.
:What I know from winning tables of other 3/6 wheels, chances for 
:exactly _one_ 3-win are only  between 20-30%, chances for a 4-win
:or more are about 15-20%.
:Adolf Muehl
:Vienna, Austria
>There was once
>a speculation that playing this wheel will reduce the
>chances of matching more than 3 on one ticket.  Any
>truth to this?
I doubt. But let's first see the table.

I think this posting contains useful hints for a successful play 
strategy! Read on!
On Thu, 9 Jan 1997 14:38:13 GMT, nveilleu@NRCan.gc.ca (Normand Veilleux)
wrote:
>consecutive drawings if you buy 1 ticket per draw.  And even after 168
>draws, there still is 0.042398 probability of having lost all draws.
That's saying 95.76 % chance of winning (>= 3) if playing the same 1 ticket
in 168 consecutive draws. IMHO an important conclusion from this would be:
 Playing the same 1 ticket in x consecutive draws is better than playing 
 x different tickets (or a wheel) in 1 draw. 
Isn't it?
If yes, then the further practical generalization of this statement 
would be:
 Don't change your numbers; ie. play always the same numbers (tickets or
 wheel) until you have a win.
 --> So one should also very well think of analysing the past draws for 
     choosing the 'right' expected numbers (it's normally a one-time task)
I think, that's it! Ie. IMO this is a very important key fact for a
successfull play strategy! Isn't it?
Sure! Because, we no longer start again from the beginning at each draw.
Instead we keep it constant since probability says "using the same 
numbers a win should occur in the next x draws..." But, if we change the
numbers each time then everything starts again from the beginning, so this
should be strictly avoided!...
What do others think on this strategy?
Coming back to the first quote above: below is the opposite case, ie. 168
different (random) tickets (or the mentioned 168 ticket wheel) in 1 draw:
>But, since 1 ticket in every 53.65514 is a winner, then 3.131107
>tickets in every 168 should also be winners.  This is the theoretical
>value obtained by simple ratio.  To prove it to yourself, all you
>have to do is take the 168 ticket wheel and count how many tickets
>win for each of the 13,983,816 possible combinations.  Add up all
>those numbers and divide by 13,983,816 to get the average.  I'm
>predicting that it will be 3.131107.
Yes, makes sense, but IMHO only on average over a long time periode.
(One needs the complete winning table for doing this calculation)
>If you do come up with the same number, then it implies that wheeling
>does not change, in any way, the average number of winning tickets.
But then also the opposite is true: wheeling is at least equal to using
the same number of any different randomly chosen single tickets. True?
Are there any situations where wheeling behaves worser than using 
randomly or even any some otherwise chosen different tickets of 
same size?

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- J. G. Kemeny, J. L. Snell, G. L. Thompson, Introduction to Fininte
  Mathematics, Second Edition, Prentice-Hall, 1956, $19, (***).
- W. Kaplan, Introduction to Analytic Functions, Addison-Wesley, 1966,
  $24, (***).
- M. C. Gemignani, Elementary Topology, Second Edition, Addison Wesley,
  1967, $22, (****).
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- P. R. Halmos, Measure Theory, Nostrand, 1950, $35, (***).
- C. L. Silver, From Symbolic Logic...to Mathematical Logic, WCB, 1994,
  $34, (!).
- J. A. Peterson, J. Hashisaki, Theory of Arithmetic, Second Edition,
  Wiley, 1963, $22, (***).
- B. V. Limaye, Functional Analysis, Halsted Press, 1981, $29, (***).
- R. E. Williamson, R. H. Crowell, H. F. Trotter, Calculus of Vector
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- G. Strang, Linear Algebra and Its Applications, Academic Press, 1976,
  $23, (***).
- J. Breuer, Introduction to the Theory of Sets, Prentice Hall, 1958,
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- R. S. Burington and C. C. Torrance, Higher Mathematics with Applications
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- E. Gaughan, Introduction to Analysis, Brooks/Cole Pub. Co.,
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- T. W. Gamelin, Uniform Algebra, Prentice Hall, 1969, $24 (***).
- J. W. Keesee, Elementary Abstract Algebra, D.C. Heath and Co.,
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- K. Knopp, Infinite Sequences and Series, Dover, 1956, $15, (***, soft
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- K. Knopp, Theory And Application Of Infinite Series, Hafner Pub. Co.,
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- R. C. Jeffrey, Formal Logic: Its Scope and Limits, McGraw Hill, 1967,
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- D. N. Clark, G. Pecelli, and R. Sacksteder, Contributions to Analysis and
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- A. Clark, Elements of Abstract Algebra, Wadsworth Publishing Co., 1971,
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- C. B. Hanneken, Introduction to Abstract Algebra, Dickenson Publishing
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- C. W. Curtis, Linear Algebra: An Introduction Approach, 2nd ed., Allyn
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- M. R. Spiegel, Applied Differential Equations, 2nd ed., Prentice Hall,
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- M. L. James, G. M. Smith, and J. C. Wolford, Applied Numerical Methods
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- G. A. Bekey and W. J. Karplus, Hybrid Computation, John Wiley & Sons,
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- F. R. Ruckdeschel, BASIC Scientific Subroutines Vol. II, McGraw Hill,
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- A. L. Edwards, Statistical Analysis for Students in Psychology and
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- P. R. Rider, An Introduction to Modern Statistical Methods, John Wiley &
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- M. Rosenblatt, Random Processes, Oxford University Press, 1962, $19 (***).
- Z. W. Birnbaum, Introduction to Probability and Mathematical Statistics,
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- R. B. Reisel, Elementary Theory of Metric Spaces, Springer-Verlag, 1982,
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- D. Moller, Ed., Advanced Simulation in Biomedicine, Springer-Verlag, 
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- D. V. Chudnovsky, G. V. Chudnovsky, H. Cohn, M. B. Nathanson, Eds., 
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- K. H. Borgwardt, The Simplex Method: A Probablistic Analysis, Springer-
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- J. R. Dias, Molecular Orbital Calculations Using Chemical Graph Theory,
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- L. S. Charlap, Bieberbach Groups and Flat Manifolds, Springer-Verlag,
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- R. L. Gue and M. E. Thomas, Mathematical Methods in Operations Research,
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- F. Hausdorff, Set Theory, 2nd ed., Chelsea Pub. Co., 1962, $29 (****).
- S. Bell, J. R. Blum, J. V. Lewis, and J. Rosenblatt, Modern University
  Calculus with Coordinate Geometry, Holden Day, 1966, $23 (****).
- P. Buser, Geometry and Spectra of Compact Riemann Surfaces, Birkhauser
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- J. L. Schiff, Normal Families, Springer Verlag, 1993, $19 (!, soft cover).
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- H. Rutishauser, Lectures on Numerical Mathematics, Birkhauswer, 1990,
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- G. F. Simmons, Differential Equations with Applications and Historical
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- P. G. Hoel, Elementary Statistics, Second Edition, Wiley, 1966, 
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I'm a quality engineer. Recently I looked production yield numbers for
the last six months. I grouped the data into 30, 25, 20...5 sample sizes
and charted results on Shewhart control charts. I was suprised to see
that the lower the sample size, the higher the yield over the time
period; the 'shape' of each curve (sample size) was approx the same. I'm
looking for a rational for this type of behavior over a range os scales.
You can respond to me 'here' or better yet:
David.Godin@mail.mei.com
thanks

In article <32d9725e.49267882@news.muenchen.org>, bm373592@muenchen.org (Uenal Mutlu) says:
>
>On Fri, 10 Jan 1997 09:22:30 -0700, Karl Schultz  wrote:
>
>>C. K. Lester wrote:
>>> 
>>> In article <32D53060.B3D@fc.hp.com>, Karl Schultz  wrote:
>>> >C. K. Lester wrote:
>>> >>
>>> >> In response to Karl Schultz's prior post,
>>> >>
>>> >> >There are no subsets.  The 168-ticket wheel will guarantee a 3-match
>>> >> >in a 6/49 lotto.
>>> >> >
>...
>>The perceived value, IMHO, is as follows.  People like to win.
>>If they can be sure to walk away with something, then they
>>might take steps to do that.  The only way to increase your
>>chances of winning is to play more numbers.  If you are
>>in the habit of playing 100+ numbers at a time and have
>>had a long losing streak, you might be inclined to play
>>the 168-ticket wheel, so that you are sure to have to make
>>that trip to the counter to claim a prize.  Actually, you
>>have a 60+% chance of getting 3 wins with 168 tickets,
>>but that is another story.  So, it is a psychological
>>thing - sure to get a win.
>
>I get a different percent value:
>
>  Since the probability for at least 3 is about 1 in 54 (to be exact
>  p=0.0186375450020), using the usual formula gives:
>
>    E = "at least once at least 3 matching nbrs in 168 consecutive 
>         draws using the same fixed 1 ticket" 
>
>    p(E) = 1 - (1 - 0.018637545)^168 = 0.9576
>
>    meaning 95.76 % of chance of occuring of the event E.
>
>(BUT: I'm still not sure if this is the same as playing 168 different 
>tickets in 1 drawing) 
>
>Which formula did you use?
>
>>In the end, you are right.  Big Deal.
>>The wheel is just a structured way to buy more
>>tickets, which, in itself will increase chances.
>
>>Now, here is a real tough question for wheel experts.
>>
>>If one plays 168 tickets using the wheel, they are sure
>>to match 3 at least once.  What does this wheel do to
>>one's chances to match more than 3???  
>
>One would need to analyse its guarantee table; I think Adolf 
>mentioned of doing this in another posting:
Here is that winning table:
That the winning table (without regard to the bonus number) of Uenal 168
wheel on 49 numbers that guarantees 3 on 6.
  6-win 5-win  4-win  3-win  blocks      %
                               1        3835601   27.4288
                               2        3058209   21.8696
                               3          821854     5.8772
                               4        3021425   21.6065
                               5+     1228555     8.7855
                      1     0-?      1769036   12.6506
                      2     0-?        139821     0.9999
                      3     0-?          60141     0.4300
                      4     0-?            4226     0.0302
                      5+   0-?            1701     0.0122
              1    0 - 7   0-?         42848     0.3064
              2    0 - 7   0-?             199      0.0014
              3    0 - 7   0-?               32      0.0002
       1   0 - 1  0 - 9   0-?           168      0.0012
Chances to win more than one 3 win are about 72 %, 

Aaron-
When you say that to determine the maximum likelihood estimate "we need to
know the universe the sample was taken from".
- exactly what do you mean?  What is a model that is approriate?
Regards-
BHActuary

Andrew Gray wrote:
>     I'm working on combining neural networks and fuzzy logic models
> with statistical techniques (regression and data reduction) for
> software metrics (for example, predicting development time based on
> the type and size of system).  While there has been a lot of work on
> neural-fuzzy, neural-genetic, fuzzy-genetic, etc. type systems I've
> only ever found a small number of researchers using AI/statistical
> techniques (presumably at least partially an indication of how few AI
> researchers follow the statistical side of things, and vice versa).
At the risk of reviving age-old threads about fuzzy logic vs.
probability, let me advise you not to bother with fuzzy logic.
There are two parts to fuzzy logic, one defensible and the other
not. The ``fuzzy'' part is one solution to the problem of
representing uncertain knowledge -- this is the defensible part.
The ``logic'' part is an attempt to reason with uncertain knowledge 
-- this is an undefensible hack.
There are (at least) two sizeable camps working in the intersection
between AI and probability/statistics. One is the neural network
crowd represented by Geoff Hinton, Radford Neal, Michael Jordan,
etc. The Neural Information Processing Systems conference is, I 
believe, the highest-quality neural networks conference -- look for
the above names in recent NIPS proceedings. Over the past several
years probabilistic and statistical viewpoints have become increasingly
important in NIPS conference papers. As discussed at length in
Warren Sarle's neural networks FAQ (sorry, I don't have pointer)
it's useful to consider neural networks as extensions of or variations
on conventional regression and classification schemes.
There's also a group associated with the Uncertainty in AI conference
which focuses on reasoning with uncertain knowledge. Their techniques
center on belief networks, which are standard probability models 
formulated in such a way that dependence or independence of variables
is represented explicitly -- this makes calculations of conditional
probabilities tractable. Judea Pearl's book _Probabilistic Reasoning
in Intelligent Systems_ is a readable, moderately technical
introduction. 
Hope this helps,
Robert Dodier
-- 
``Ainda nos faz lembrar os belos tempos'' -- on the prow of a fishing
boat.

AaCBrown (aacbrown@aol.com) wrote:
: bhactuary@aol.com (BHActuary) in
: <19970109161200.LAA11015@ladder01.news.aol.com> asks:
: > Does anyone have insight into the maximum likelihood
: > estimate and prediction error from a trading rule (or any rule)
: > which has produced a profit on historical data.  Invariably,
: > if you apply such a rule to future data you will get less profit
: > (and sometimes even a loss)!
: This is really a selection bias problem. If you select a trading rule that
: did well in the past it will tend to do less well in the future. This is
: true of many things. The baseball player with the best batting average by
: the all-star break will probably have a lower average in the second half
: of the season. The children of the tallest person in the world will
: probably be shorter than he is.
<< deleted...>>
I think  "selection bias"  is the proper name, when the selection 
formula does something to recommend it.  However, companies touting
stocks have also been guilty of gross 'over-fitting' - 
finding formulas that fit because there are relatively few events
to be modeled, and a lot of parameters to model them with, so, 
technically, one runs out of degrees of freedom, and practically a 
perfect fit occurs by chance.  Those fail because there was never
any good to them in the first place.
Finally - if someone does find a sane formula for betting the stock
market, it should only work so long as it is basically secret.  I
mean, it is like betting on NFL football:  twenty-five years ago,
there was a time you could get fat, by betting on the home-team
underdogs;  the home-field was being underrated.  But, once the word
of that analysis got out, the betting lines shifted.  SO it was no
longer true.
Maybe this is a macro-economic aspect of the Heisenberg Principle ?
Rich Ulrich, biostatistician                wpilib+@pitt.edu
http://www.pitt.edu/~wpilib/index.html   Univ. of Pittsburgh

Uenal Mutlu wrote:
> 
> On Fri, 10 Jan 1997 09:22:30 -0700, Karl Schultz  wrote:
> 
> >C. K. Lester wrote:
> >>
> >> In article <32D53060.B3D@fc.hp.com>, Karl Schultz  wrote:
> >> >C. K. Lester wrote:
> >> >>
> >> >> In response to Karl Schultz's prior post,
> >> >>
> >> >> >There are no subsets.  The 168-ticket wheel will guarantee a 3-match
> >> >> >in a 6/49 lotto.
> >> >> >
> ...
> >The perceived value, IMHO, is as follows.  People like to win.
> >If they can be sure to walk away with something, then they
> >might take steps to do that.  The only way to increase your
> >chances of winning is to play more numbers.  If you are
> >in the habit of playing 100+ numbers at a time and have
> >had a long losing streak, you might be inclined to play
> >the 168-ticket wheel, so that you are sure to have to make
> >that trip to the counter to claim a prize.  Actually, you
> >have a 60+% chance of getting 3 wins with 168 tickets,
> >but that is another story.  So, it is a psychological
> >thing - sure to get a win.
> 
> I get a different percent value:
It is "perceived" value.  That intangible value/advantage
that people THINK they are getting from a wheel.
I was discussing psychology more than math.
> 
>   Since the probability for at least 3 is about 1 in 54 (to be exact
>   p=0.0186375450020), using the usual formula gives:
> 
>     E = "at least once at least 3 matching nbrs in 168 consecutive
>          draws using the same fixed 1 ticket"
> 
>     p(E) = 1 - (1 - 0.018637545)^168 = 0.9576
> 
>     meaning 95.76 % of chance of occuring of the event E.
> 
This is right if you pick the 168 tickets randomly.
If you use the wheel, you get 100%.
I am not a wheel expert, but if you pick 168 tickets randomly,
you are going to have some number of duplicate coverage for 3-matches.
All the wheel does is
reduce the number of redundant 3-match coverages and makes sure that
all possible 6 number combinations result in a 3-match.  
There is some redundancy left over.
Here is the output of Richard Lloyd's wheel checker.
Overall 3-match combination coverage summary:
Covered 10 times :     1
Covered  6 times :     3
Covered  5 times :     6
Covered  4 times :    59
Covered  3 times :    60
Covered   twice  :    52
Covered   once   :  2782
Covered   never  : 15461
Total:      2963 / 18424 (16.1%)
    Combination      Covered  Elapsed   Speed   To Go
 1 12 13 14 15 16    1210362    0:03   403454    0:31
 2  3  4  5  6  7    1712304    0:03   570768    0:21
[Reporting switched off during the final 30 seconds]
     Finished       13983816    0:26   537839    0:00
Losing  combinations:        0 (  0.0%)
Winning combinations: 13983816 (100.0%)
Total   combinations: 13983816 (for a 6 from 49 lottery)
> (BUT: I'm still not sure if this is the same as playing 168 different
> tickets in 1 drawing)
(different issue - start a new thread?)

In article <32D66CC6.25A@fc.hp.com>, Karl Schultz  wrote:
>> NO NO NO... sheesh almighty. I was referring to the "perceived value" of such
>> a scheme... as in, "what value is buying 168 tickets for a guaranteed
>> three-match?" Maybe I should have said, "Big deal."
>
>The above representation of your question is much better than the
>vague "So what?".
I agree. :)
>The perceived value, IMHO, is as follows.  People like to win.
..
>but that is another story.  So, it is a psychological
>thing - sure to get a win.
What it is, basically, is, "You are guaranteed to lose AT MOST $165. You 
could lose even less!"
hehe That's not very good, in my perception.
>Now, here is a real tough question for wheel experts.
>
>If one plays 168 tickets using the wheel, they are sure
>to match 3 at least once.  What does this wheel do to
>one's chances to match more than 3???  There was once
>a speculation that playing this wheel will reduce the
>chances of matching more than 3 on one ticket.  Any
>truth to this?
I would think that since any set has an equal chance of winning, that there is 
no reduction/addition to one's chance to match a certain number...
Right?
Later!
ck

>It would be useful if we had a simulation software which for example 
LOTSIM - Simulation-Program for all pick-X type Lottery Games
Ok, a very first version of the mentioned LOTSIM Simulation Program for 
Lotto is ready. Email me if anyone wants to try it out (warning: IMHO 
useful only for mathematically interessted people; runs as a commandline 
program in a DOS-Box under Win95 or WinNT. ZIPed approx. 60 KB, EXE only).
(BTW, Karl Schultz posted a similar C-program some days ago in r.g.l.) 
Below are 2 runs of simulations. The difference is: in the first run
fixed ticket numbers were used during all draws in each of the trials, 
whereas in the second simulation, in each draw the ticket was randomly 
chosen. 
The values of the first are slightly better, but should this difference be 
really significant? IMHO yes, but I'm not yet sure if the differences are 
also statistically significant.
It seems that fixed tickets give a higher degree of success than random 
tickets. Can this be true, or is the difference not really significant?
(Any comments from statisticians? Which significance testing method would 
be appropriate for this?)
*******************************************************************
LOTSIM v1.00beta - Math Simulations for Lotto 6/49, 6/54, 5/32 etc.
Author/(c): U.Mutlu (bm373592@muenchen.org)
Limits    : vMax=54 kMax=7 
Tested    : with 6/49 under Win95
Usage     : LOTSIM cTrials cDraws cWin v k fTicketsFixed ...
Cmdline   : LOTSIM 25000 54 3 49 6 1 ...
Simulation settings:
 v=49 k=6 cDraws=54 cTicketsPlayedPerDraw=1 fTicketsFixed=1
 cTrials=25000 
 A ticket is defined as having 6 different nbrs 1..49
 Draw numbers are generated by the standard RNG, ie. the rand()
 function. Seed (srand(time)) is done once at pgmstart.
 Bonus Number not drawn and not calculated
 A 'win' is defined as having >= 3 matching on a ticket
Simulation results:
  wins occurance      %       cumul%
  ---- --------- --------- ---------
 >= 10         0   0.00000   0.00000
     9         0   0.00000   0.00000
     8         0   0.00000   0.00000
     7         1   0.00400   0.00400
     6        10   0.04000   0.04400
     5        68   0.27200   0.31600
     4       356   1.42400   1.74000
     3      1556   6.22400   7.96400
     2      4745  18.98000  26.94400
     1      9261  37.04400  63.98800
     0      9003  36.01200 100.00000
       ---------
           25000
Interpretation:
 25000 trials were made, each consisted of 54 drawings and in 
 each drawing 1 fixed ticket was played. Simulations were done 
 for Lotto 6/49. A win is defined as having >= 3 matching numbers.
 The table says for example "after 25000 trials of 54 drawings
 each, there were 9261 cases where only 1 draw in each series
 (ie. of ea. 54) had a win". And, we're dealing with "at least" wins.
*******************************************************************
LOTSIM v1.00beta - Math Simulations for Lotto 6/49, 6/54, 5/32 etc.
Author/(c): U.Mutlu (bm373592@muenchen.org)
Limits    : vMax=54 kMax=7 
Tested    : with 6/49 under Win95
Usage     : LOTSIM cTrials cDraws cWin v k fTicketsFixed ...
Cmdline   : LOTSIM 25000 54 3 49 6 0 ...
Simulation settings:
 v=49 k=6 cDraws=54 cTicketsPlayedPerDraw=1 fTicketsFixed=0
 cTrials=25000 
 A ticket is defined as having 6 different nbrs 1..49
 Draw numbers are generated by the standard RNG, ie. the rand()
 function. Seed (srand(time)) is done once at pgmstart.
 Bonus Number not drawn and not calculated
 A 'win' is defined as having >= 3 matching on a ticket
Simulation results:
  wins occurance      %       cumul%
  ---- --------- --------- ---------
 >= 10         0   0.00000   0.00000
     9         0   0.00000   0.00000
     8         0   0.00000   0.00000
     7         0   0.00000   0.00000
     6        12   0.04800   0.04800
     5        67   0.26800   0.31600
     4       366   1.46400   1.78000
     3      1522   6.08800   7.86800
     2      4729  18.91600  26.78400
     1      9289  37.15600  63.94000
     0      9015  36.06000 100.00000
       ---------
           25000
Interpretation:
 25000 trials were made, each consisted of 54 drawings and in 
 each drawing 1 random ticket was played. Simulations were done 
 for Lotto 6/49. A win is defined as having >= 3 matching numbers.
 The table says for example "after 25000 trials of 54 drawings
 each, there were 9289 cases where only 1 draw in each series
 (ie. of ea. 54) had a win". And, we're dealing with "at least" wins.
-- Uenal Mutlu (bm373592@muenchen.org)   
   * Math Research * Designs/Codes * SW-Development C/C++ * Consulting * 
   Loc: Istanbul/Turkey + Munich/Germany