Welcome Guest
Log In | Register )
You last visited November 28, 2014, 11:36 am
All times shown are
Eastern Time (GMT-5:00)

The Powerplay Option

Topic closed. 32 replies. Last post 2 years ago by jimmy4164.

Page 2 of 3
51
PrintE-mailLink

United States
Member #124498
March 14, 2012
7024 Posts
Offline
Posted: February 7, 2013, 12:22 pm - IP Logged

There is no good random generated by computers following articles on Internet. If true would be the random from random.org, then I would want to play big cash money against it. Maybe there are a few patents that generate better random results, but then again, what is random? It surely doesn't have to follow any mathematical rule. That is what it is.

Main Entry:1ran£dom Pronunciation:*ran-d*m Function:noun Etymology:Middle English, succession, surge, from Anglo-French randun, from Old French randir to run, of Germanic origin; akin to Old High German rinnan to run — more at  RUN Date:1561

 : a haphazard course   –at random : without definite aim, direction, rule, or method  *subjects chosen at random*

Main Entry:random walk Function:noun Date:1905

 : a process (as Brownian motion or genetic drift) consisting of a sequence of steps (as movements or changes in gene frequency) each of whose characteristics (as magnitude and direction) is determined by chance

    SergeM's avatar - alas
    Economy class
    Belgium
    Member #123705
    February 27, 2012
    2924 Posts
    Offline
    Posted: February 7, 2013, 9:29 pm - IP Logged

    Runen

      jimmy4164's avatar - andy warhol.jpg
      State of Mind
      United States
      Member #93949
      July 10, 2010
      2177 Posts
      Offline
      Posted: February 8, 2013, 12:21 am - IP Logged
      Random sampling
      Referring to a simple Lottery "1 of 5". Suppose the numbers 1, 2, 3, 4, 5, dropped to 20 circulations with frequencies of 2, 2, 4, 8, 4. To create a table that presents the data in a usable form.
      _ N _ _ F _ _ RF _ _ _ CRF
      1 2 0.10 0.10
      2 2 0.10 0.20
      3 0.20 0.40 4
      0.80 0.40 8 4
      4 5 1.00 0.20
      How can you choose the numbers to select the number of probability is directly proportional to its frequency of occurrence in history (or, equivalent, is equal to its relative frequency)? A very simple way would be to take your tiles set of 10 ornaments China, with pen on one of them to call a number one, the other two, two, three, four, and two 5:0 pm 4. Then put them in a jar, shake well, and blindly remove nodules. Probability, which would you choose 4 number dominoes is 0.40, the probability that you select a number of dominoes is 0.10, and so on. All nodes of the fingers of choice (the next election the junta extracted must be returned to the Bank, and that the unrest) is random with probability of the frequency selected on the number fell in previous editions. This-the random selection, but weighted random (probability-weight), with "hot" items weighing more than the weight of "cold" numbers. Otherwise you could do the same random sample, using uniform random numbers, and the following distribution function. Our distribution function:
      _N_ _CRF_
      1 0.10
      2 0.20
      3 0.40
      4 0.80
      5 1.00
      Use RAND () to select one of two-digit uniform random number. Assume that this is-the number 0.34. Look at the column of CRF. The number fell between 0.34 and 0.40 0.20, choose the number 3 (You randomly selects 3 in distributors). Use RAND () to select another random number. Suppose that this time you have 0.69. From 0.69-between 0.40 and 0.80, select the number 4. And so on. What is happening here? It is very simple. Since the RAND () produces a uniform random number, then the probability that the number of falls-equal to 0.20 -0.40 0.20 (less 0.20 0.40), then there is the relative frequency of fall-3 emission (see the table at the top of this page.) Thus, the choice to interpret 0.34, a number between 0.20 and 0.40, which is similar to a random selection of numbers 3. Likewise, our random number is between 0.40 and 0.80 0.69, is interpreted as a randomly selected number 4:. The relative frequency corresponding to 4, 0.40, is the same as the probability that we could choose a uniform random number between 0.40 and 0.80 This random selection process can be automated (no conscience), using the following simple rules. Rule. Use RAND () to select a random number ravnomernoraspredelennoe, call it r. If R falls between two values of a column, the FRC, choice of column N is a number that corresponds to the higher of the two values of which obtained r. If R is not equal to one of the values in the column, column choice no CRF corresponding number. Illustration with the rule. If R = 0.53, picked 4 because 0.53-lies between 0.40 and 0.80, and 4 correspond to 0.80. If R = 0.31, choose 3 because it is between 0.20 0.31 and 0.40, and 3 corresponds to 0.40. If R = 0.20, choose 2. If R = 0.80, select 4. Lottery conducted by the Government as Lotto Texas use mechanisms of mixing and playing balls, trying to ensure that they totally random occurrence. By "completely random" I mean simply that each ball in a game full of balls, can be selected with equal probability in any quantity. This is a sample function of a uniform distribution. If, in fact, the machine reaches the perfect objective randomization, the RAND function (50) to the Texas Lotto or RAND (40) for the Plus Lotto-how a good a way as any to select the numbers for your tickets.

      Your weighting scheme using RAND() should provide you with sets more or less congruous with those in your input.  Unfortunately, I suspect your reason for wanting to do this is firmly rooted in a belief in the Gambler's Fallacy.  Unless the objects being selected are clearly not equal in weight, shape, or size, the value of your output will be of lttle or no value.

        Avatar
        bgonçalves
        Brasil
        Member #92566
        June 9, 2010
        1549 Posts
        Offline
        Posted: February 8, 2013, 6:43 am - IP Logged
          jimmy4164's avatar - andy warhol.jpg
          State of Mind
          United States
          Member #93949
          July 10, 2010
          2177 Posts
          Offline
          Posted: February 8, 2013, 12:20 pm - IP Logged

          Thanks Dr San,

          That is a very interesting article.  (Trying To Be Random in Selecting Numbers for Lotto - Boland) It could be subtitled, "Where Psychology Meets Mathematics!"

          I wasn't able to access Speckman's article, "Lottery Loophole Explained."  Do you know what the loophole was, and how long after 1986 it took them to close it?

          Here are a couple quotes from Boland's article that others here might enjoy:

          "For Lotto 6/42, P(MG = 1) = 0.56 is the probability of a selection containing two consecutive numbers. Students usually find it very surprising and nonintuitive that it is more likely than not that a random selection will contain two consecutive numbers. Before giving students the probability of two consecutive numbers appearing in Lotto, it is a worthwhile exercise to test their intuition by asking them for estimates of it!"  (Read the article if this intrigues you!)

          "5. Conclusions

          "21 The whole concept of randomness is a delicate one, and one about which considerable research has been done, particularly in the field of psychology. Reichenbach (1949) claimed that humans are unable to produce a random sequence of responses, even when explicitly asked to do so, and considerable research since then (including our work) generally supports this. Many teachers/lecturers have told us of classroom activities they use to get students thinking about randomness, and Green (1997) gives an interesting account of an experiment on recognizing randomness.

          "22 How well can individuals perform when they attempt to be random generators for a game like Lotto? Some interesting insights into human intuition about randomness may be obtained by asking a class of students to participate in an exercise that tries to answer this question. We asked each of the students in a large class to act once as a random generator for the winning numbers in our Irish National Lottery - Lotto 6/42 game. The results were then analysed and compared with actual recent winning selections in our Lotto game, as well as with another set simulated by computer. Using boxplots, histograms, QQ-plots, and some basic measures of spread in a sample, one is able to generate interesting classroom discussions about biases that individuals seem to possess. We observed (perhaps not surprisingly) that there seems to be a propensity for individuals to select numbers in increasing order. We also observed that individuals tend to select numbers which on the average are smaller (by at least two) than would be expected from a truly random generator. Of course, in other countries or states where a different form of Lotto is played, the results will probably differ. When it comes to the spread in a selection, we observed that individuals tend to make selections that are reasonably spread out as measured by sample variance, but not in other ways (for example, as measured by the so-called minimum gap in a selection). In particular, we observed a clear reluctance of individuals (compared to a truly random generator) to make selections containing consecutive numbers." 

          --Jimmy4164

          P.S.  Remember:  Você não pode tirar sangue de um nabo!  Smile

            SergeM's avatar - alas
            Economy class
            Belgium
            Member #123705
            February 27, 2012
            2924 Posts
            Offline
            Posted: February 8, 2013, 12:38 pm - IP Logged

            Even knowing the probability you will have hard to guess right based on that.

            Let's say that you have a pick 5 game with 56 numbers. You think, hope, guess, dreamt, ... that there will be exactly 2 consecutive numbers drawn. 5-2=3 not consecutive numbers, but that is still unsure, you can have 5 consecutive numbers or just none.

            Again, 2 consecutive numbers of 56 numbers, 55 tickets to get 2/5 right!

              jimmy4164's avatar - andy warhol.jpg
              State of Mind
              United States
              Member #93949
              July 10, 2010
              2177 Posts
              Offline
              Posted: February 8, 2013, 1:40 pm - IP Logged

              It is actually worse than that, SergeM.

              Don't forget that the probability of 2 consecutives is quite a ways from 1.0, so your 55 tickets don't even guarantee that!

              By the way, I agree with your signature line!  And if any of the cigarette corporation officials who suppressed laboratory evidence of the hazards of smoking as far back as the 1930s are still alive, they should be prosecuted, and hopefully, jailed!

              --Jimmy4164

                SergeM's avatar - alas
                Economy class
                Belgium
                Member #123705
                February 27, 2012
                2924 Posts
                Offline
                Posted: February 8, 2013, 2:07 pm - IP Logged

                You can prosecute and jail people working in the tobacco industry today. Same for resellers and mean smokers. People with master degrees are working in the tobacco industry, they should be lynched.

                  Avatar
                  bgonçalves
                  Brasil
                  Member #92566
                  June 9, 2010
                  1549 Posts
                  Offline
                  Posted: February 8, 2013, 2:14 pm - IP Logged
                  Hello, this condition of sergem followed can also put vertically
                  Example =
                  01 02 03 04 05
                  06 07 08 09 10
                  11 12 ...
                  Depending on how it assembles an array, but this example = 01.06. Are consecutive numbers
                  Another example = 05.10, are numbers in a row vertically, there are also the delta value for the pair worth = two and three example = 01.03 = value delta 2 and delta value 04 01.04
                  So in addition to the sequel in horizonta, = we have in vertical sequences 01.02 (as array) 01.06 and caom delta value 2.3,
                    Boney526's avatar - NjlpLogo
                    New Jersey
                    United States
                    Member #99034
                    October 18, 2010
                    1439 Posts
                    Offline
                    Posted: February 8, 2013, 2:21 pm - IP Logged

                    Coin Toss,

                    Incredible!  Isn't it? Smile

                    I got lucky last night after only letting it run a couple of hours and hit a 5+0.  Set to Powerplay, the $2M hit still only pushed the ROI to about $0.90.  Without the Powerplay, it was significantly less.  Which brings me to my reason for this thread:

                    Obviousely, if you hit a 5+0 or 5+1 within your lifetime you're going to be ahead of the game, but is the higher ROI with Powerplay (without a Jackpot) worth the tradeoff of having less chances at the big ones?  I'm thinking it's a personal decision based on your beliefs.  If you really believe you can engineer a big hit, you'll probably opt out of the Powerplay, otherwise, you'll opt in to take advatage of the [approx] 60% increase in ROI for all hits of 4+1 and below.

                    What do you think?

                    --Jimmy4164

                    I disagree.  Your ROI may be higher, but your overall expense per ticket is also greater.  If it's .30 and .50 return on the dollar, but you can spend $2 or$3 dollars respectively, the overall cost is larger playing the Powerplay option.  And if you play when there are large jackpots only, the EV actually probably get's larger without the powerplay option.  I didn't actually do the math to find out - I'm kinda just guessing because I don't wanna do the work right now (and b/c I don't really like to play Powerball anyway.)

                      Avatar
                      bgonçalves
                      Brasil
                      Member #92566
                      June 9, 2010
                      1549 Posts
                      Offline
                      Posted: February 8, 2013, 2:49 pm - IP Logged

                      I hope it will be inspiring for other lottery players.  List contains also few terms related to gambling, like "combinadic".

                       

                       

                       "Classical" views: http://en.wikipedia.org/wiki/Classical_definition_of_probability http://en.wikipedia.org/wiki/Frequency_probability

                       Alternative points of view: http://en.wikipedia.org/wiki/Probability_interpretations

                      http://en.wikipedia.org/wiki/Empirical_probability  "Empirical probability, also known as relative frequency, or experimental probability,  is the ratio of the number favorable outcomes to the total number of trials, not in  a sample space but in an actual sequence of experiments. In a more general sense,  empirical probability estimates probabilities from experience and observation.  The phrase a posteriori probability has also been used an alternative to empirical  probability or relative frequency."

                       Bayesian probability: http://en.wikipedia.org/wiki/Bayesian_probability  "According to the Bayesian probability calculus, the probability of a hypothesis  given the data (the posterior) is proportional to the product of the likelihood  times the prior probability (often just called the prior). The likelihood brings  in the effect of the data, while the prior specifies the belief in the hypothesis  before the data was observed."

                      http://en.wikipedia.org/wiki/Propensity_probability  "Theorists who adopt this interpretation think of probability as a physical propensity,  or disposition, or tendency of a given type of physical situation to yield an outcome  of a certain kind, or to yield a long run relative frequency of such an outcome."

                      http://en.wikipedia.org/wiki/Algorithmically_random_sequence  "An algorithmically random sequence (or random sequence) is an infinite sequence  of binary digits that appears random to any algorithm."

                       Please, see http://en.wikipedia.org/wiki/Algorithmically_random_sequence#Relative_randomness

                      http://en.wikipedia.org/wiki/Kolmogorov_complexity  "In algorithmic information theory (a subfield of computer science), the  Kolmogorov complexity (also known as descriptive complexity,  Kolmogorov-Chaitin complexity, stochastic complexity, algorithmic  entropy, or program-size complexity) of an object such as a piece of  text is a measure of the computational resources needed to specify the  object."

                      http://en.wikipedia.org/wiki/Pseudorandomness  "A pseudo random process is a process that appears random but is not.  Pseudo random sequences typically exhibit statistical randomness while  being generated by an entirely deterministic causal process."

                      http://en.wikipedia.org/wiki/Statistical_randomness  "A numeric sequence is said to be statistically random when it contains  no recognizable patterns or regularities; sequences such as the results  of an ideal die roll, or the digits of Pi exhibit statistical  randomness. Statistical randomness does not necessarily imply "true"  randomness, i.e., objective unpredictability."

                      http://en.wikipedia.org/wiki/Ramsey_theory  "Ramsey theory, named after Frank P. Ramsey, is a branch of mathematics  that studies the conditions under which order must appear. Problems in  Ramsey theory typically ask a question of the form: how many elements of  some structure must there be to guarantee that a particular property  will hold?"

                      http://www.ciphersbyritter.com/RES/RANDTEST.HTM  Excellent randomness literature survey compiled by Terry Ritter. Highly recommended.

                       See also unpublished PhD dissertation by Michiel van Lambalgen on randomness: http://staff.science.uva.nl/~michiell/docs/fFDiss.pdf

                       This document is very interesting, because it refers to intuitive opinion of children:  "Children’s understanding of randomness as a model" http://www.bsrlm.org.uk/IPs/ip28-3/BSRLM-IP-28-3-09.pdf  Definitely worth reading!

                       "PEOPLE’S INTUITIONS ABOUT RANDOMNESS AND PROBABILITY" http://tinyurl.com/mvz8m5  "A recent empirical study indicates that students in introductory  statistics class are generally confused about the different notions of  probability (Albert, 2003). Clearly, continuing to teach only the  frequentist conception cannot reduce the confusion. This implies to  students either that there is only one “correct” conception of  probability or that the frequentist and Bayesian conceptions are  competitive, which should not be the case (Vranas, 2001). Moreover, an  exclusive focus on frequentist notions may conflict with the students’  intuitions and representations about probability (see e.g., Hawkins and  Kapadia, 1984). In any case, as emphasized by Konold (1991, p. 144),  “the teacher cannot, by decree, enforce a normative view.”

                      http://en.wikipedia.org/wiki/Calculus_of_predispositions  "According to Aron Katsenelinboigen, calculus of predispositions is  another method of computing probability. Both methods may lead to the  same results and, thus, can be interchangeable. However, it is not  always possible to interchange them since computing via frequencies  requires availability of statistics, possibility to gather the data as  well as having the knowledge of the extent to which one can interlink  the system’s constituent elements. Also, no statistics can be obtained  on unique events and, naturally, in such cases the calculus of  predispositions becomes the only option."

                      http://en.wikipedia.org/wiki/Outlier  "In statistics, an outlier is an observation that is numerically distant  from the rest of the data. They can occur by chance in any distribution,  but they are often indicative either of measurement error or that the  population has a heavy-tailed distribution. In the former case one  wishes to discard them or use statistics that are robust to outliers,  while in the latter case they indicate that the distribution has high  kurtosis and that one should be very cautious in using tool or  intuitions that assume a normal distribution. A frequent cause of  outliers is a mixture of two distributions, which may be two distinct  sub-populations, or may indicate "correct trial" versus "measurement  error"; this is modeled by a mixture model."

                      http://en.wikipedia.org/wiki/Imprecise_probability  "The notion of Imprecise probability is used as a generic term to cover  all mathematical models which measure chance or uncertainty without  sharp numerical probabilities. It includes both qualitative (comparative  probability, partial preference orderings,...) and quantitative modes  (interval probabilities, possibility theory, belief functions, upper and  lower previsions, upper and lower probabilities, ...). Imprecise  probability models are needed in inference problems where the relevant  information is scarce, vague or conflicting, and in decision problems  where preferences may also be incomplete. Imprecise Probability Theory  aims not to replace, but to complement and enlarge the classical notion  of Bayesian probability, approach to probability theory, by providing it  with tools to work with weaker information states."

                      http://en.wikipedia.org/wiki/Combinadic  "In mathematics, a combinadic is an ordered integer partition, or composition.  Combinadics provide a lexicographical index for combinations. Applications for combinadics  include software testing, sampling, quality control, and the analysis of gambling games."

                      http://en.wikipedia.org/wiki/Factoradic  "In combinatorics, factoradic is a specially constructed number system.  Factoradics provide a lexicographical index for permutations"

                      http://www.designinference.com/documents/2002.09.rndmnsbydes.pdf  "Randomness by Design", William A. Dembski

                       Informative and inspiring materials from Alan Hajek: http://philrsss.anu.edu.au/people-defaults/alanh/  I recommend "Fifteen Arguments Against Hypothetical Frequentism"

                       Another interesting web page, summarizes different views on probability: http://www.geocities.com/potential_continuity/physicalprobability.html

                       "From Algorithmic to Subjective Randomness" Thomas L. Griffiths, Joshua B. Tenenbaum http://cocosci.berkeley.edu/tom/papers/algrand.pdf

                       "Probability, algorithmic complexity, and subjective randomness" http://web.mit.edu/cocosci/Papers/complex.pdf

                       "Interpretations of Probability" http://plato.stanford.edu/entries/probability-interpret/

                       Terence Tao: "Structure and randomness in combinatorics"  (http://arxiv.org/abs/0707.4269)  "Combinatorics, like computer science, often has to deal  with large objects of unspecified (or unusable) structure.  One powerful way to deal with such an arbitrary object is to  decompose it into more usable components. In particular,  it has proven profitable to decompose such objects into a  structured component, a pseudo-random component, and a  small component (i.e. an error term); in many cases it is the  structured component which then dominates." _________________

                        jimmy4164's avatar - andy warhol.jpg
                        State of Mind
                        United States
                        Member #93949
                        July 10, 2010
                        2177 Posts
                        Offline
                        Posted: February 9, 2013, 2:54 am - IP Logged

                        dr san,

                        Thanks for the bibliography.

                        --Jimmy4164

                          jimmy4164's avatar - andy warhol.jpg
                          State of Mind
                          United States
                          Member #93949
                          July 10, 2010
                          2177 Posts
                          Offline
                          Posted: February 9, 2013, 3:13 am - IP Logged

                          I disagree.  Your ROI may be higher, but your overall expense per ticket is also greater.  If it's .30 and .50 return on the dollar, but you can spend $2 or$3 dollars respectively, the overall cost is larger playing the Powerplay option.  And if you play when there are large jackpots only, the EV actually probably get's larger without the powerplay option.  I didn't actually do the math to find out - I'm kinda just guessing because I don't wanna do the work right now (and b/c I don't really like to play Powerball anyway.)

                          Boney526,

                          Please double check my calculations. If they're correct, it seems to me that the $3 Powerplay is a better bet than the $2 ticket, especially if you ignore the 5+0 and 5+1 Jackpots.  It's still not much consolation though.  I wonder how many people would play Roulette if the payouts there were structured in proportion to these?  Smile

                          As usual, choosing a payout value for the 5+1 is problematic. If they really do pay back 0.50, an adjustment would need to be made there. It's surprising that the commission would allow for such a significant difference between the 2 options, about 10.6 cents, not including a 5+1 hit.

                          While running the Powerball Simulator, notice how the 2 highlighted numbers below dominate as long as you don't get a 5+0 or 5+1 hit.

                                  $2 Powerball             $3 Powerplay
                          Hit Odds           Payout      ROI*  Payout      ROI*

                           PB 1/55           4          .036   12         .073
                          1+1 1/111          4          .018   12         .036
                          2+1 1/706          7          .005   14         .007
                          3+0 1/360          7          .010   14         .013
                          3+1 1/12,245       100        .004   200        .005
                          4+0 1/19,088       100        .003   200        .003
                          4+1 1/648,976      10,000     .008   40,000     .021

                          Third Tier Expected ROI       .084              .158

                          5+0 1/5,153,633    1,000,000  .097   2,000,000  .129
                          5+1 1/175,223,510  40,000,000 .114   40,000,000 .076

                          Total Expected ROI            .295              .363

                          * Expected Return For Each Dollar Spent Purchasing Tickets

                          --Jimmy4164

                            Boney526's avatar - NjlpLogo
                            New Jersey
                            United States
                            Member #99034
                            October 18, 2010
                            1439 Posts
                            Offline
                            Posted: February 10, 2013, 11:19 pm - IP Logged

                            The ROIs might be close (if you include the top two prizes, which I'd argue you should,) but you are increasing the total expense.  If you don't count those - then Powerplay is a better bet, but both are terrible without the top two prizes inclusion (and are pretty terrible, anyway.)

                             

                            And as the jackpot grows, I'd imagine the ROI of not taking powerplay to exceed the ROI of taking it, although, with a starting jackpot taking Powerplay is better from an ROI standpoint.

                             

                            If I was forced to play with or without, I still pick without.

                              jimmy4164's avatar - andy warhol.jpg
                              State of Mind
                              United States
                              Member #93949
                              July 10, 2010
                              2177 Posts
                              Offline
                              Posted: February 11, 2013, 2:41 am - IP Logged

                              The ROIs might be close (if you include the top two prizes, which I'd argue you should,) but you are increasing the total expense.  If you don't count those - then Powerplay is a better bet, but both are terrible without the top two prizes inclusion (and are pretty terrible, anyway.)

                               

                              And as the jackpot grows, I'd imagine the ROI of not taking powerplay to exceed the ROI of taking it, although, with a starting jackpot taking Powerplay is better from an ROI standpoint.

                               

                              If I was forced to play with or without, I still pick without.

                              It looks like the "break even" point is a Jackpot of $112M.

                              The total ROI, NOT including the 5+1 Jackpot, for

                              PB vs PP   is  .181 vs .287.  (See above)

                              So, excluding the 5+1 Jackpot, the ROI for Powerplay is significantly better, nearly 60% better.

                              Increasing the Jackpot to $112M changes the

                              PB (Jackpot Only) ROI to .320 and the

                              PP (Jackpot Only) ROI to .213 . 

                              .181 + .320 = .501  (PB)

                              .287 + .213 = .500  (PP)

                              .500 is the amount most lotteries claim they give back, so I think this is a reasonable conclusion.  As the Jackpot increases above $112M,  PB will outperform PP, in total, as you indicated.

                              Admittedly, neither is a good bet when compared to 93% and better returns possible at casinos, but I'm surprised you would pass up the 60% advantage of the Powerplay 2nd tier and lower to get those extra shots at the [very remote] Jackpot.