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Wavepack

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Feb 2, 2023, 12:30 pm
A Chi-Squared Goodness of Fit computes the probability that a drawing sample came from a uniform probability distribution. The results:
- Powerball: Probability of new white ball draws from a non-uniform pdf = 0.651666 (# draws = 521)
- Mega Millions: Probability of white ball drawn from a non-uniform pdf = 0.758665 (# draws = 546)
  Mega Millions: Probability of MegaBall drawn from a non-uniform pdf = 0.206510 (# draws = 546)
This indicates a possibility that information exists in past draws that can improve pick prediction.

The formula:

pdf = probability distribution function
W = Chi-Squared Goodness of fit.

W has a chi-squared pdf with M-1 degrees of freedom
N = number of draws
X(k) = Count of number times ball k is chosen over D draws, k = [1..M]
M = Number of different balls
m = Number of balls picked without replacement at each draw

W = [M*(M-1)*sum_k{X(k)^2 - (m^2 N^2)/M}] / (M - m)*N*m
cottoneyedjoe

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Feb 2, 2023, 2:35 pm

Good post, Wavepack. Several years ago I did this very analysis on a variety of mechanical ball drawing lotteries and computerized pseudo-random number generator lotteries. I tested various time frames too.

In the end, I identified only three PRNG lotteries in the US that exhibited statistically significant and on-going non-uniform behavior. But ultimately the flaws I found were not severe enough that you could exploit them for profit. At most, you could shave a little off the odds by avoiding certain numbers and patterns.

Of all the mechanical ball lotteries, I didn't find any that had statistically significant biases.

Good luck to you!
Wavepack

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Feb 2, 2023, 3:33 pm

Yeah, that's what I am finding too. Hardly any information that can be used to improve estimation.

However, I did find a little something (maybe), emphasis on little. Using Bayesean optimal estimation with 1st and 2nd order statistics, I did find one method with the following white ball [0W 1W 2W 3W 4W 5W] performance for Powerball new white balls.

Percent Total Matches = [59.064327 33.918129 5.263158 1.754386 0.000000 0.000000], NumTests = 171

Expected Percent Matches for Powerball white balls = [67.84%, 28.27%, 3.7%, 0.179% ep ep]

Not statistically significant, but it was the best performance I saw in terms of payoff. Three 3W hits over 171 tests.

Another method had 11% 2W hits and one 3W hit over 121 tests.

Percent Total Matches = [62.809917 24.793388 11.570248 0.826446 0.000000 0.000000], NumTests = 121

Still losing money hand over fist, unless you are one of the lucky 5 or 6 ball winners, which is super rare.
cottoneyedjoe

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Feb 2, 2023, 7:21 pm

Quote: Originally posted by Wavepack on Feb 2, 2023
Yeah, that's what I am finding too. Hardly any information that can be used to improve estimation.

However, I did find a little something (maybe), emphasis on little. Using Bayesean optimal estimation with 1st and 2nd order statistics, I did find one method with the following white ball [0W 1W 2W 3W 4W 5W] performance for Powerball new white balls.

Percent Total Matches = [59.064327 33.918129 5.263158 1.754386 0.000000 0.000000], NumTests = 171

Expected Percent Matches for Powerball white balls = [67.84%, 28.27%, 3.7%, 0.179% ep ep]

Not statistically significant, but it was the best performance I saw in terms of payoff. Three 3W hits over 171 tests.

Another method had 11% 2W hits and one 3W hit over 121 tests.

Percent Total Matches = [62.809917 24.793388 11.570248 0.826446 0.000000 0.000000], NumTests = 121

Still losing money hand over fist, unless you are one of the lucky 5 or 6 ball winners, which is super rare.

Another issue is that even if you successfully identify a lottery with an exploitable weakness, it gets patched up pretty quickly -- either when the physical balls or machinery become deprecated and replaced, or when the PRNG machine undergoes scheduled routine service and updates. Funny how an innocuous software update can completely change the behavior of a PRNG.

The best anomaly that I found in a PRNG lottery several years ago just up and disappeared when there was a leadership change in the state lottery agency and a large staff shakeup. I'm guessing the new people who where hired decided to make their mark by replacing the computers. Since then, all statistical tests of the draw data show no statistically significant deviations from uniformity. Bummer.
Wavepack

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Feb 3, 2023, 12:25 am

Quote: Originally posted by cottoneyedjoe on Feb 2, 2023
Another issue is that even if you successfully identify a lottery with an exploitable weakness, it gets patched up pretty quickly -- either when the physical balls or machinery become deprecated and replaced, or when the PRNG machine undergoes scheduled routine service and updates. Funny how an innocuous software update can completely change the behavior of a PRNG.

The best anomaly that I found in a PRNG lottery several years ago just up and disappeared when there was a leadership change in the state lottery agency and a large staff shakeup. I'm guessing the new people who where hired decided to make their mark by replacing the computers. Since then, all statistical tests of the draw data show no statistically significant deviations from uniformity. Bummer.

Yeah, the time variance of the statistics. They switch ball sets constantly, swap out individual balls when out of spec, etc. I have no model information for that time variance to incorporate into the estimation. Absent of a model, I account indirectly for time variant statistics with two time length parameters. During the discovery phase, which I'm still in, I adjust those parameters manually. Once I get a better feel of what information is available to help estimation, I can automate that parameter adjustment to maximize payout. Baby steps first during the exploratory phase.

The error most commonly made these days when implementing a PRNG to simulate randomness is to not seed the PRNG with enough bits of entropy. In earlier times, the PRNG chosen might of had built-in weaknesses beyond a weak seed, but the standard ones included in libraries today are often good PRNGs. In my State, they seed the PRNG from noise derived from quantum level randomness from nuclear decay. But they had a major system flub in 2017 when their hardware failed and they seeded the PRNG with zero for multiple days in a row!!! Same numbers picked on daily lottos!

I totally agree with the Lottery Post website creator (Todd) that computer picked lottos are too vulnerable to attack (because of the myriad ways that computers can be hacked, including computers not connected to a computer network). I do not play PRNG lotteries or any other computer picked lotteries because the State-run lotteries have done zero to convince me that their computer system is not vulnerable to attack. At least with the Powerball and Mega Millions, they show video evidence of a highly chaotic system that is highly random during the draw.

If the lottery carries out ball screening before each draw as they claim, and broadcast the draw live, and encrypt and digitally sign ticket data with time stamps before the draw, then reasonable evidence will have been provided to the players that the lottery draw is fair. Unfortunately, they do not show all of that evidence to the players, so the players are unsure if the lotto is fair. Suspicions of rigging are reasonable if the lottos don't provide all the evidence that proves the draw and ticket creation is on the up and up
cottoneyedjoe

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Feb 3, 2023, 11:38 am

Quote: Originally posted by Wavepack on Feb 3, 2023
Yeah, the time variance of the statistics. They switch ball sets constantly, swap out individual balls when out of spec, etc. I have no model information for that time variance to incorporate into the estimation. Absent of a model, I account indirectly for time variant statistics with two time length parameters. During the discovery phase, which I'm still in, I adjust those parameters manually. Once I get a better feel of what information is available to help estimation, I can automate that parameter adjustment to maximize payout. Baby steps first during the exploratory phase.

The error most commonly made these days when implementing a PRNG to simulate randomness is to not seed the PRNG with enough bits of entropy. In earlier times, the PRNG chosen might of had built-in weaknesses beyond a weak seed, but the standard ones included in libraries today are often good PRNGs. In my State, they seed the PRNG from noise derived from quantum level randomness from nuclear decay. But they had a major system flub in 2017 when their hardware failed and they seeded the PRNG with zero for multiple days in a row!!! Same numbers picked on daily lottos!

I totally agree with the Lottery Post website creator (Todd) that computer picked lottos are too vulnerable to attack (because of the myriad ways that computers can be hacked, including computers not connected to a computer network). I do not play PRNG lotteries or any other computer picked lotteries because the State-run lotteries have done zero to convince me that their computer system is not vulnerable to attack. At least with the Powerball and Mega Millions, they show video evidence of a highly chaotic system that is highly random during the draw.

If the lottery carries out ball screening before each draw as they claim, and broadcast the draw live, and encrypt and digitally sign ticket data with time stamps before the draw, then reasonable evidence will have been provided to the players that the lottery draw is fair. Unfortunately, they do not show all of that evidence to the players, so the players are unsure if the lotto is fair. Suspicions of rigging are reasonable if the lottos don't provide all the evidence that proves the draw and ticket creation is on the up and up

Interesting! California Lottery's PRNG used to involve randomly choosing between a hardware RNG (one that derived random numbers from atmospheric noise or nuclear decay or some other physical process) and a garden-variety PRNG algorithm. On any given day the daily numbers could have come from the piece of hardware or an algorithm. I don't know if they still do it that way, they removed a lot of information from the website during a revamp a couple years ago. For example, now you can't even get more than 6 month's worth of draw data, whereas before you could download it going back decades.
Wavepack

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Feb 3, 2023, 12:35 pm

Quote: Originally posted by cottoneyedjoe on Feb 3, 2023
Interesting! California Lottery's PRNG used to involve randomly choosing between a hardware RNG (one that derived random numbers from atmospheric noise or nuclear decay or some other physical process) and a garden-variety PRNG algorithm. On any given day the daily numbers could have come from the piece of hardware or an algorithm. I don't know if they still do it that way, they removed a lot of information from the website during a revamp a couple years ago. For example, now you can't even get more than 6 month's worth of draw data, whereas before you could download it going back decades.

Texas helps to keep the Powerball and Mega Millions honest by publishing the lotto draw results going back to 2015 or so. That eases independent analysis by the public. They preserve pick order, which is important for analysis. The Arizona Lotto also stinks when it comes to publishing draw results. Only six months back, position information not recorded, and it's not available for file download (cvs or text would be sufficient).

State-run lottos screw over the public big time by keeping winning ticket payouts of tickets not claimed within six months. What a rip off. Super greedy State-run lotteries taking legitimate lottery prize wins from the public. If the lotto ads were accurate, they'd include a miserly grinch swiping back winning tickets from the public. No prize for you! A coercive monopoly State lottery abusing their coercive monopoly status. It's coercive in the sense that they forcefully prevent anyone else from running a lottery, thus securing monopoly status and all the bad service typically associated with coercively enforced monopolies.

The AZ lotto screwed me out of a Powerball winning ticket (PB hit only, $5 over $3 ticket cost) earlier this year. They have proof that I claimed the ticket because I entered the unique number associated with the ticket into the AZ Players website. I called, pointing out they have that proof. They refused to honor the payout even with full proof in their own database. The call was well before their BS 6 month time out period.
Wavepack

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Feb 7, 2023, 2:58 pm

I also implemented 3rd order Bayesean optimal estimation, despite the limited data set.

I also implemented a blend between 1st and 2nd order statistics.

I also tried joint Bayesean optimal estimation of intra-draw pairs and inter-draw pairs, which you could view as a subset of full 3rd order statistics.

I have developed a program that picks in a way to maximize 3W, 4W, and 5W hits since that's what pays. My analysis shows that there exists a tiny bit of information in past draw data that can improve expected payout in the long run over uniform random picks. To extract it, you need graduate level probability and statistics, plus the time and skill to develop and debug analysis code.

The optimal strategy with Powerball and Mega Millions is to not play, which is what I do when the jackpot is < 300M. For funsies, I play when the jackpots are large (> 300M). When I do play, I will post my picks here before the draw Since both lotteries had recent jackpot hits, it will be at least a month before I post any predictions.
Orange71

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Mar 1, 2023, 2:46 pm

IMHO, in general government-run lotteries (in the "western world" at least) are not going to be exploitable in terms of statistical frequency attack. A lot of time, energy and money(!) is obviously put into trying. Eventually, if you look at enough lotteries, you'll find "improbable" outcomes and what appears to be "non-randomness". Highly likely, that doesn't mean there is a flaw. Just if you sample enough, you get unusual patterns. (It's inevitable.) The people who have successfully exploited lotteries in the past did so due to flawed prize structures, where jackpots not won fall through to lower prize tiers. Not a statistical frequency attack on specific numbers appearing or not. There's even a recent movie about the Midwest couple who pulled this off.

This doesn't mean you can do nothing at all. Two actions make sense to me:

(1) Pick numbers above 30 or 31. People are known to pick ages and birthdays, so months and days under 12 and 31 respectively. Therefore, not doing what these people do in theory means that for a pari-mutuel prize you hit, there is statistically lower expected value of other winners splitting the pool. I haven't seen lotteries that publish number frequencies of tickets bought, but perhaps some do.

(2) Randomly pick numbers such that there aren't repeats across successive plays, or as few as possible repeats depending on how many tickets you buy. This will probably have minimal effect on the Expected Value of your return, but it will reduce your Standard Deviation. Extreme example: Suppose we have a 6/54 Lotto (happens to be the structure of my state's game). I manually pick two tickets, but don't follow my own maxim: (a) 32 37 39 46 47 53 and (b) 32 37 39 46 47 51. If the winning numbers are 32 37 39 46 47 54, I'm now a 2x winner in the 5/6 pari-mutuel pool. Suppose there are two other winners in that pool. Had I picked 32 37 39 46 47 only on one ticket I would have received 33.3% of the pool's prize. Instead I get 50% because of the two matching tickets. So I paid double (i.e. bought another ticket) and didn't get double my return.

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