Randomness of the lottery

[anxious]

Hello fellow gamblers..

I hope we have a statistician among our kind because i need to test the randomness of my local lottery.

The numbers are drawn by a mechanical device. Hence, it is possible that there is some bias in the drawn numbers due to fluctuations of the weights of the balls, or by the fact that the balls enter the bowl in a certain order.

Specifically, I d like to know the following:

1) Does the marginal frequency distribution of numbers drawn correspond to the one expected under randomness (i.e., are the numbers drawn in a truly random way in the sense that the probability that each number is drawn is 1 over the number of balls in the bowl)? and is the distribution of numbers drawn uniform over the columns on the form.

2) Do the mean and variance of the sum of the numbers, the distribution of odd and even numbers, and the distribution of the minimum and maximum correspond to those expected under randomness ?

3) Does the distribution of waiting times (the number of complete draws before a number is drawn again) correspond to the one expected under randomness, and does the Lotto have a memory ( Do numbers that have come up in a certain draw, return in the next draw)?

if y'all have a software in mind that will yield the results i look for, i d appreciate if you advise it to me..

best regards
anxious

 

[GillesD]

Randomness of a lottery

anxious

I do believe that most loteries are quite random, not perfectly so (this by the definition of randomness itself) but quite near it. I also believe that all agencies running loteries are looking at all the data generated to ensure that no bias exists and that variuos methods are taken for that reason (multiple sets of balls, different mixing times, etc.).

Yes, you can check if the actual results are near the theorical values that you could expect. Theory on probabilities are well documented or you can count how many combinations meet a certain criteria (let's say 6 consecutive numbers for a 6/49 lottery). In a post a few days back (in Actual draws with a low probability of occurring), I mentionned one method for such calculations but many sites wil show all this information. And obviously, as the number of draws increses, you should be nearer the theorical values.

I maintain a database with quite a few statistics like those you mentionned and most results are not far from the actual theorical values. Up to now, with the Canadian Lotto 6/49, the only major difference I have found is that higher numbers have come out slightly but significantly more often (on a statistical basis) compared to lower numbers. But in a normal system, perfection (even for randomness) is not often obtained. I would not bet on getting 500 heads and 500 tails in 1000 coin flips. You may get this a few times if you repeat the experiment quite a few times but more likely anywhere from 450 to 550, with most results being with ±10.

 

[anxious]

I do believe that most loteries are quite random, not perfectly so (this by the definition of randomness itself) but quite near it. I also believe that all agencies running loteries are looking at all the data generated to ensure that no bias exists and that variuos methods are taken for that reason (multiple sets of balls, different mixing times, etc.).

GillesD,

what you say is what anyone would assume.. Yet, i am one suspicious kind of native..hehe.

For a long time i have been asking myself. Self, how can it be that every Christmas and New Years eve the jackpots reach outrageous proportions while in the rest of the year Jackpots are much, much less??

GillesD, if i play a game that is supposed to be random i d like to verify that it is so myself. Simply, I dont trust lottery agencies. There is a lot of scam going on these days to do so.

anxious

 

[Rob50]

Anxious, only a quick question: What would be the reason the lottery agencies would be interested to have "non-random" draws, and even if they would have a hidden interest, haw in your mind this translate to "a lot of scam going on these days"?

 

[anxious]

rob50, thank you for the naive question..

Lottery agencies in the land of the natives, where i come from are mostly govermental agencies.. Now, a percentage of the money from the jackpots benefits a number of other activities and organizations. Not all money gathered from bets go toward the next jackpot. Therefore the lottery agencies have a clear interest toward 1) creating larger jackpots 2) creating more interest for the game by advertizing large jackpots.
Some of the other organizations benefiting from the lottery are athletic organizations (a variety of athletic teams ranging from soccer to track and field), Media companies(TV advertizments), Government emploees etc..

Hence, it is not entirely out of this world for one to assume that the lottery draws are rigged. That the balls are magnetized in such a way as to create jackpots..

The "a lot of scam going on these days" relates to what is being revealed these days on TV, in the land of the natives, about governmental agancies corruption and mismanagement..

regards
anxious

 

[GillesD]

Rigged loteries

anxious

You are right in saying that there is a lot of money involved in loteries and that a lot of money is going to good and worthy causes. I think that you must have contributed some money to the pot of gold (I know I have, not a lot but over the years it adds up).

The Canadian loteries have been around for more than 25 years and this has involved quite a few men and women, and this is true also for the United States and European nations (not counting all other ones). I find it rather odd that something (rigging loteries) has been going on for so long with so many people without coming out and well documented in the press. That would make such a nice scandal. But do not tell me the press can be silenced.

The other point that troubles me is: Why kill the golden goose? I know that. at least in Québec and in Canada in general, the lotery agencies are generating a lot of money (basicly a tax but a lotery is better sounding). And that's why mayor Drapeau created the "taxe volontaire" in Montreal in the 1970's. Now think of the drop in revenues (5%,25%, 50% and I would say even more) if an actual case of rigging was well publicized. Would you put your $2 worth in a lottery?, I know I would not. And THIS I think is what keeps them honest.

 

[anxious]

But do not tell me the press can be silenced.

Oh, this is exactly what i am going to tell you GillesD.. I dont know about the lotteries in other lands, but the Lottery Agency in the Land of the Natives, is the biggest spender of advertising money both on TV and in the written press. Tribal leaders rely on this money for the reasons I outlined earlier. No one is going to pick a fight with the lottery agency. Believe me.

Also, which one of the lowly Natives is able to prove that the lottery they play does not follow the laws of randomness??

Having analyzed my draws, having selected the most likely numbers to hit and having set my 20 number net (wheel), I got the most unlikely number 1 to hit. This tells me that the lottery people did not want to give the money away but rather create another Jackpot.

Folks, would you be playing poker with marked cards?

All I am saying is that any gambler that respects himself should know the kind of game he is beting his money on..

anxious

 

[Rob50]

Anxious, thanks for the effort in responding my naive question with that an astute answer. One thing though, all you have said is common knowledge and unfortunately (or fortunately) I don't see any logical link to why the lottery agencies would be willing to rig the lottery for the reasons you have mentioned. The risk they would take this way would be devastating for themselves, as very correctly has been pointed out bu GillesD. Maybe not some natives you mention here, but the worst student of statistics would be able to demonstrate in the simplest way if a lottery were rigged. Also, allow me to very friendly tell you that the illusion you might have about your ability to predict the likely numbers to hit cannot be a proof of rigged lotteries just because numbers that you consider unlikely to hit are drawn.
Sincerely
Rob 50

 

[anxious]

..but the worst student of statistics would be able to demonstrate in the simplest way if a lottery were rigged.

Rob50, you sound like you can help me verify that my lottery follows the laws of randomness and that all i worry about is an illusion in my mind. After all this is what i am after in this thread.
A set of statistical tests that when applied to a set of historical drawings will demonstrate that the lottery in not biased.

I thank you in advance
anxious

 

[Rob50]

Well, I can give it a try, just let me know to which lottery are you refering.

 

[GillesD]

Bias or not

anxious

One way to determine if results are biased is to do significance tests in order to conclude if the observed frequencies for a given statistics differ significantly from the expected frequencies that are calculated.

Based on actual results and expected results, you build up contigeny tables for a given confidence level (such as 95% or 99%). This is what I maintain in a Excel file for 14 statistics on the Canadian Lotto 6/49. Here is an example with the difference between the highest number and the lowest one in a winning combination.

The first column gives the difference between the last and first number, the second the actual result and the third the theorical value expected:
- 05-15: 020 -- 018
- 16-20: 057 -- 065
- 21-25: 162 -- 165
- 26-30: 303 -- 317
- 31-35: 483 -- 491
- 36-40: 611 -- 616
- 41-48: 761 -- 725

In this case, it fairly obvious that there is not much difference between the two sets of data (for 2397 draws). And a contigency table (at a 95% confidence level) confirms that actual values do not differ significantly from expected values. And as a matter of fact, the difference is also not significant even at the 99% level.

But, from this, you can not conclude that the lottery is not rigged since you can always manipulate data to get what you want. So it is back to square one.

 

[anxious]

..In this case, it fairly obvious that there is not much difference between the two sets of data (for 2397 draws).

Indeed GillesD.. Not much difference there.. Surprising !!

2 Questions:
1) How do you calculate expected frequencies (the number of times we expect a number x to occur in n drawings should be n times the hypergeometric probability)
2) The 2397 draws you tested must span a large period of time. Within this period there must have been draw-machine changes. Would that affect your calculations or you take this into account?

I dont think that the Lottery in the Land-of-the-Natives will exibit such uniformity as your Lottery..

Can you post this excel file to use with my own data?

Well, I can give it a try, just let me know to which lottery are you refering.

Rob50, i am sorry, i don't wish at this point to reveal the Land-of-the-Natives.
If you have a piece of software that will yield me results i d be happy to peruse..

anxious

 

[GillesD]

Randomness of draws

anxious

I will try to answer your questions with some examples:

1 -Expected frequencies:

Expected frequencies can calculated with laws of probabilities. Rolling dice, you have 1 chance on 6 to get a 6 (16.7%), and 1 chances in 12 to roll 2x6 (9.33%) if you roll 2 dices.

It is the same thing in lotteries; you just have to calculate the number of combinations that meet your criteria and divide it by the total number of combinations (13,983,816 different combinations for a 6/49 lotery). For 3 odd numbers and 3 even numbers, there are 4,655,200 possible combinations, so you have 33.29% chances of getting this type of combination, and this independantly of all previous draws. In Excel, the formula is: =COMBIN(25,3)*COMBIN(24,3)

Of course, calculating some probability is sometime a little more complicated and in many case, it is easier to run through all combinations and count how many combinations meet one or more criteria.

2 - Variations in systems:

The Canadian Lotte 6/49 started in June 1982 with one draw per week and went 2 draws per week in September 1985.

Of course, lottery agencies have many machines available (they can not allow one machine to fail when is it time to draw the winning combination). Also they have many sets of balls available and these are tested by an independant lab to verify that each ball is very similar, but it is impossible to have 49 balls prefectly identical in weight, roundess, resilience and other properties. A while back (way back), I posted a link for an article from the Standard Council of Canada on testing related to lotteries.

In my case, different machines and sets of balls do not affect my calculations as I strongly believe that winning is a matter of luck. Some have it, some do not. And I rely on the same set of numbers since a while back but no big gain.

 

[anxious]

Meanwhile, and while we talk about whether the lottery follows the laws of Randomness, i am losing big time at the 5/45 plus 1 bonus..

For one more time, I stretched my 20,5,3,4 wheel out, to only hit (for one more time) a meager 3 numbers..

But over time, i kind of understand whats happening..
a) The lottery authority collects all bets from the players
b) They check which ball they have to magnetize in order to generate one more jackpot
c) they do it.

It is my feeling that unless one defines the word "Randomness" no meaningful conclusion can be drawn whether a statistic is indicative of randomness or not.

So, what if the frequency of a lottery is what it is? Is this frequency picture what characterizes Randomness in the first place? This just says that by the laws of nature and over time all numbers must show up. Just because everything else in Gods kingdom exhibits such a high degree of symmetry the lottery cannot be the exception. That’s a great and respectable conclusion but too general.

As far as i know, no one can define Randomness satisfactorily.. It is one of those intractable problems of our time that wait for a solution by some future generation. For if someone had the precise definition of Randomness it would amount to being able to consistently beat the lottery, as well.

anxious

 

[Rob50]

Well, for one thing, the lottery is the last and the least subject related to the "Randomness", though it is true that maybe it is the most popular subject. Now dear anxious you are not going to solve the problem of "Randmomness" posting some thing, or reading in this lotto-forum, and also no matter how much money you are gonna lose, whether the lottery authorities make up the jack-pots, or keep them appear "randomly". If you seriously want to know more about "randomness", may be it is time to start reading some good books authored by real authorities in the field (and I mean in probability and statistics), and not little popular brochures about how to hit the jack-pot. I don't like and don't want to be rude, but if you are not really interested in that, better play your numbers and don't talk about things that you think are to be defined yet. Second, reading a couple of good books on the subject (again, not charlatan's books) will make it clear that hitting the jack-pot is just a matter of luck! Sincerely Rob.

 

[anxious]

Well.. I was hoping for some healthy reaction in this thread..

Now dear anxious you are not going to solve the problem of "Randomness" posting something, or reading in this lotto-forum

Dear Rob50, you never know.. Internet is a big place.. Maybe some alien out there will read this thread and decides to give me what I am looking for. And this is a tool to be able to tell whether my lottery is biased and if it is in what way..

..better play your numbers and don't talk about things that you think are to be defined yet. Second, reading a couple of good books on the subject (again, not charlatan's books) will make it clear that hitting the jack-pot is just a matter of luck!

Would you call Niederreiter, L’Ecuyer, Matsumoto, Hellekalek, Knuth, Marsaglia, Fishman etc. charlatans?? Anyway, which some good books according to you?
What i gather from your text though is this:
1) you know something and got offended by the ignorance of this native
2) you think that you know something which may prove to be nothing (or very little)
3) you want to say something but you are not saying it

Bottom line.. I have asked you this before. Can you provide a set of statistical tests that when applied to a set of historical drawings will demonstrate that the lottery is or is not biased and in what way?

sincerely anxious

 

[Rob50]

Dear anxious,

It was absolutely not my intention to discourage you from trying, or hoping to solve the problem of “Randomness”. At best I just wanted to save efforts in a direction that always have been proved vain and wrong. It is absolutely your right to dream about aliens providing the solution. I apologize for trying to kill a dream, which even if never becomes true keeps always the dreamer happy. Sorry, again.
I did not mention any charlatan’s name. It was you that came up with names. Still, if any of them somehow, somewhere has asserted that the lottery is breakable, I would not hesitate for a second to call him that way. Nevertheless, I don’t think, from what I have read from Niederreiter, or Hellekalek, that they had any ideas about breaking lotteries (!). As for the others, Knuth is a computer programmer, so I don’t see why do you include him in this list. As for the others I don’t think they would have anything to do with what we are discussing. Having said that, it really amazes me that how it is possible that a guy like you (I don’t see as essential the fact that you are native, or like to pass for one), after having read Niederreiter (I assume you have) still thinks or hopes or dreams, or something.
As for your numbered list:
1) Yes, I know something but I did not get offended by the ignorance of that native. I just thought to help him a little bit.
2) I am sure that whatever I know is very, very little. One thing I am sure though, it is GOOD.
3) I have always said what I wanted to say.

Bottom line, what you have asked me, you could have asked Niederreiter. He has answered your question. You only need to extract from his books what you need about testing “how random” are the numbers in a sequence of pretended to be random numbers, or pseudo-random numbers.

Sincerely Rob