The Powerplay Option

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

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.

link=http://www.amstat.org/publications/jse/secure/v7n3/boland.cfm#ziemba

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! 

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!

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

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.

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.)

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." _________________

dr san,

Thanks for the bibliography.

--Jimmy4164

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? 

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

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.