Chi-Squared Goodness of Fit Test for Lotteries

Reply #8

Orange71 — Wed, 01 Mar 2023 19:46:17 GMT

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 lotte... [ More ]

Reply #7

Wavepack — Tue, 07 Feb 2023 19:58:00 GMT

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... [ More ]

Reply #6

Wavepack — Fri, 03 Feb 2023 17:35:15 GMT

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 win... [ More ]

Reply #5

cottoneyedjoe — Fri, 03 Feb 2023 16:38:01 GMT

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 m... [ More ]

Reply #4

Wavepack — Fri, 03 Feb 2023 05:25:26 GMT

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 para... [ More ]

Reply #3

cottoneyedjoe — Fri, 03 Feb 2023 00:21:39 GMT

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

Reply #2

Wavepack — Thu, 02 Feb 2023 20:33:16 GMT

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... [ More ]

Reply #1

cottoneyedjoe — Thu, 02 Feb 2023 19:35:30 GMT

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 avoidi... [ More ]

Chi-Squared Goodness of Fit Test for Lotteries

Wavepack — Thu, 02 Feb 2023 17:30:52 GMT

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 dr... [ More ]