Lotto: Statistics question

[andyandy]

Then, run a series of 208 trials and see what you'll get on average.

What you will get is a lower probability of win - right?

no no no no no no no.

CF why are you continuing to argue? Maths isn't like other topics where one can play with sophistry until the opposing poster grows tired of the games....

in the OP example

1)It does not affect your odds of winning.
2)It will have an effect upon your EV (though given the tendency for people to over pick 1-9, a skew which favours 10+ may actually be beneficial dependent on the stats)

that is it. Many many people have now said the same thing. It is incontrovertible.

 

[Macoy]

You computed it wrong. 42,072,307,200 is the number of possible sequences that could be drawn. But you need the number of possible sets. In other words, when matching the winning and guessed combination, your calculation compares the numbers one by one, assuming they are ordered, when they are in fact unordered: 3, 7, 13, 18, 22, 31, 35 actually matches 22, 7, 35, 13, 3, 18, 31.

If you want to go all the way, you need to divide 42,072,307,200 with the number of ways that 7 numbers can be permutated, or reordered while remaining the same set, and that is 7! = 5040. Therefore, the correct number of possible results is 8,347,680, which is what other people got.



I have no idea how you arrived to that result, and it is wrong. No glitch in the ticket generator can affect the probability of winning. Other people have extensively explained why. It also follows from the simple realization that in a truly random lottery, there is no way whatsoever to alter the probability of winning. Any calculation or reasoning that suggests otherwise is necessarily wrong.

You are absolutely correct - I am completely wrong. Ahh well...

 

[drkitten]

Then, run a series of 208 trials and see what you'll get on average.

What you will get is a lower probability of win - right?

Wrong

If he runs a series of 208 trials, he will get a probablity of winning somewhere "around" 0.166666.

If he runs several series of 208 trials, he will get several different probabilities of winning. Some of them will be higher, some of them will be lower. They will average "around" 0.166666

If he runs a lot of series of 208 trials, he will get a lot of different probabilities that will average "around" 0.166666.

In each case, I could define "around" more precisely in terms like "the value will be between X and Y with probability .95" if it were that important. But it isn't.

 

[CFLarsen]

no no no no no no no.

CF why are you continuing to argue? Maths isn't like other topics where one can play with sophistry until the opposing poster grows tired of the games....

in the OP example

1)It does not affect your odds of winning.
2)It will have an effect upon your EV (though given the tendency for people to over pick 1-9, a skew which favours 10+ may actually be beneficial dependent on the stats)

that is it. Many many people have now said the same thing. It is incontrovertible.

You seem to be under the impression that I am arguing against 1 and 2. I'm not.

 

[ben m]

According to the article, the expected pay out is not the same for the group.

Not the person. The group.

CFLarsen, please explain *exactly* what you think is going on. What is the odds-of-winning of a "fairly vended" ticket? Of a biased one? What is the expected cash value of a fairly vended ticket? Of a biased one? What is the lottery-service's weekly expected payout if they vend fairly, or if they vend unfairly? (Note: this depends somewhat on what happens to unwon jackpots. 100 people buying "the same number" increases the odds that the prize will go unwon; if the money rolls over to the next week, then the lottery service's average weekly payout does not depend on how they vend their tickets.)

Then, run a series of 208 trials and see what you'll get on average.

What you will get is a lower probability of win - right?

No. If you think you can find a pattern in the data below, please tell us, not just the pattern, but your estimate of its *statistical significance*. These are the same 4 RNGs I described in an earlier post.

fair, no-1, some-1, only-1
208 trials, run 1: 0.163462 0.158654 0.134615 0.163462
208 trials, run 2: 0.158654 0.134615 0.115385 0.158654
208 trials, run 3: 0.149038 0.168269 0.129808 0.163462
208 trials, run 4: 0.192308 0.168269 0.168269 0.192308
208 trials, run 5: 0.197115 0.168269 0.163462 0.1875
208 trials, run 6: 0.0865385 0.1875 0.139423 0.153846
208 trials, run 7: 0.149038 0.211538 0.163462 0.192308
208 trials, run 8: 0.1875 0.221154 0.105769 0.177885
208 trials, run 9: 0.139423 0.168269 0.177885 0.134615
208 trials, run 10: 0.158654 0.177885 0.134615 0.182692
208 trials, run 11: 0.158654 0.134615 0.158654 0.149038
208 trials, run 12: 0.144231 0.192308 0.173077 0.144231
208 trials, run 13: 0.197115 0.182692 0.192308 0.168269
208 trials, run 14: 0.192308 0.182692 0.1875 0.149038
208 trials, run 15: 0.211538 0.173077 0.163462 0.149038
208 trials, run 16: 0.201923 0.201923 0.177885 0.1875
208 trials, run 17: 0.1875 0.1875 0.134615 0.134615
208 trials, run 18: 0.177885 0.216346 0.182692 0.192308
208 trials, run 19: 0.1875 0.173077 0.153846 0.139423
208 trials, run 20: 0.0865385 0.192308 0.1875 0.158654
208 trials, run 21: 0.153846 0.211538 0.129808 0.158654
208 trials, run 22: 0.149038 0.125 0.129808 0.153846
208 trials, run 23: 0.206731 0.129808 0.173077 0.211538
208 trials, run 24: 0.144231 0.158654 0.177885 0.201923
208 trials, run 25: 0.182692 0.168269 0.144231 0.216346
208 trials, run 26: 0.192308 0.240385 0.139423 0.158654
208 trials, run 27: 0.139423 0.163462 0.245192 0.129808
208 trials, run 28: 0.163462 0.206731 0.163462 0.163462
208 trials, run 29: 0.192308 0.168269 0.134615 0.163462
208 trials, run 30: 0.158654 0.153846 0.134615 0.168269
208 trials, run 31: 0.153846 0.149038 0.216346 0.129808
208 trials, run 32: 0.1875 0.177885 0.197115 0.1875
208 trials, run 33: 0.125 0.206731 0.221154 0.149038
208 trials, run 34: 0.158654 0.149038 0.201923 0.173077
208 trials, run 35: 0.125 0.134615 0.1875 0.1875
208 trials, run 36: 0.158654 0.173077 0.158654 0.216346
208 trials, run 37: 0.182692 0.1875 0.168269 0.1875
208 trials, run 38: 0.129808 0.168269 0.211538 0.192308
208 trials, run 39: 0.153846 0.134615 0.173077 0.206731

If you want more data I can PM you as much as you like.

 

[PenguinWarrior]

It may be "fair" if we look at the single gambler in the very long run, but it isn't "fair" to the group of people who got "heads" selected for them.

Why not? They have a 50% chance of winning, just as those who have the tails tickets do. This is because the method of selecting the result (the coin) is unbiased. Since the probabilities of any particular outcome occurring (in this case heads or tails, with the lottery any one selection of numbers) are the same, the biased selection doesn't alter your chance of winning (since it gives you one out of a set of equally likely outcomes and, by definition, any one of a set of equally likely outcomes is exactly as likely to occur as any of the others, it mattersnot a jot how you came to select it).

Now what it will effect is the distribution of winnings across the various possibilities (since the distribution is not now uniform, some sets of numbers are more likely to be selected than others, so these are more likely now to be selected by two or more people, and thus will be more likely to produce lower payouts - assuming that the amount won decreases with the number of winners - a very safe assumption with a lottery). However, I suspect that the effect on big wins will be very small (since so few people have all the numbers on any one week anyway), and that the effect was much smaller than the skew produced by human tendencies (I.e. picking birthdays, not picking groups of numbers close together etc), and of course those who pick the numbers not favoured by the machine will now be more likely to get a bigger payout, so the average winnings won't change across the whole group of consumers.

 

[Dan O.]

The links I have seen are all in Danish. We don't know the exact difference, but you can see statistics here:

danskespil.dk

Click on "Spil". Next level, click on "LOTTO". Next level, click on "LOTTO". Sidebar, click "Statistik".

Is there an online translator that can translate this language? The page looks like it's just showing numbers picked that were winners.

There seams to be a significant skew at the top end but I don't know if this is a skew in how often the number 36 comes up as a winner or if the players just have some aversion to playing the number 36.

 

[ben m]

Aha, and, yes, one more thing:

If you run the dice lottery (pick 1-6) and two people play *random numbers*, then you indeed expect a winner (i.e., one or more winners) in 11/36ths of your weekly drawings. If you run the dice lottery and two people play *always different numbers*, you expect a winner in 12/36ths of your weekly drawings. If two people always play the *same number*, you expect a winner in only 1/6th of the drawings. Is that what you were getting at?

Whether this is important or not depends on the size of the lottery. If the mean number of winners is above 1, then adding more people to the payout has only a tiny affect on the weekly payout probability. If the mean number of winners is < 1, adding more people to the same numbers reduces the weekly payout probability linearly.

 

[CFLarsen]

CFLarsen, please explain *exactly* what you think is going on. What is the odds-of-winning of a "fairly vended" ticket? Of a biased one? What is the expected cash value of a fairly vended ticket? Of a biased one? What is the lottery-service's weekly expected payout if they vend fairly, or if they vend unfairly? (Note: this depends somewhat on what happens to unwon jackpots. 100 people buying "the same number" increases the odds that the prize will go unwon; if the money rolls over to the next week, then the lottery service's average weekly payout does not depend on how they vend their tickets.)

We don't know how much bias the flawed RNG had.

No. If you think you can find a pattern in the data below, please tell us, not just the pattern, but your estimate of its *statistical significance*. These are the same 4 RNGs I described in an earlier post.

fair, no-1, some-1, only-1
208 trials, run 1: 0.163462 0.158654 0.134615 0.163462
208 trials, run 2: 0.158654 0.134615 0.115385 0.158654
208 trials, run 3: 0.149038 0.168269 0.129808 0.163462
208 trials, run 4: 0.192308 0.168269 0.168269 0.192308
208 trials, run 5: 0.197115 0.168269 0.163462 0.1875
208 trials, run 6: 0.0865385 0.1875 0.139423 0.153846
208 trials, run 7: 0.149038 0.211538 0.163462 0.192308
208 trials, run 8: 0.1875 0.221154 0.105769 0.177885
208 trials, run 9: 0.139423 0.168269 0.177885 0.134615
208 trials, run 10: 0.158654 0.177885 0.134615 0.182692
208 trials, run 11: 0.158654 0.134615 0.158654 0.149038
208 trials, run 12: 0.144231 0.192308 0.173077 0.144231
208 trials, run 13: 0.197115 0.182692 0.192308 0.168269
208 trials, run 14: 0.192308 0.182692 0.1875 0.149038
208 trials, run 15: 0.211538 0.173077 0.163462 0.149038
208 trials, run 16: 0.201923 0.201923 0.177885 0.1875
208 trials, run 17: 0.1875 0.1875 0.134615 0.134615
208 trials, run 18: 0.177885 0.216346 0.182692 0.192308
208 trials, run 19: 0.1875 0.173077 0.153846 0.139423
208 trials, run 20: 0.0865385 0.192308 0.1875 0.158654
208 trials, run 21: 0.153846 0.211538 0.129808 0.158654
208 trials, run 22: 0.149038 0.125 0.129808 0.153846
208 trials, run 23: 0.206731 0.129808 0.173077 0.211538
208 trials, run 24: 0.144231 0.158654 0.177885 0.201923
208 trials, run 25: 0.182692 0.168269 0.144231 0.216346
208 trials, run 26: 0.192308 0.240385 0.139423 0.158654
208 trials, run 27: 0.139423 0.163462 0.245192 0.129808
208 trials, run 28: 0.163462 0.206731 0.163462 0.163462
208 trials, run 29: 0.192308 0.168269 0.134615 0.163462
208 trials, run 30: 0.158654 0.153846 0.134615 0.168269
208 trials, run 31: 0.153846 0.149038 0.216346 0.129808
208 trials, run 32: 0.1875 0.177885 0.197115 0.1875
208 trials, run 33: 0.125 0.206731 0.221154 0.149038
208 trials, run 34: 0.158654 0.149038 0.201923 0.173077
208 trials, run 35: 0.125 0.134615 0.1875 0.1875
208 trials, run 36: 0.158654 0.173077 0.158654 0.216346
208 trials, run 37: 0.182692 0.1875 0.168269 0.1875
208 trials, run 38: 0.129808 0.168269 0.211538 0.192308
208 trials, run 39: 0.153846 0.134615 0.173077 0.206731

If you want more data I can PM you as much as you like.

It would be more interesting to see the program.

Why not? They have a 50% chance of winning, just as those who have the tails tickets do. This is because the method of selecting the result (the coin) is unbiased. Since the probabilities of any particular outcome occurring (in this case heads or tails, with the lottery any one selection of numbers) are the same, the biased selection doesn't alter your chance of winning (since it gives you one out of a set of equally likely outcomes and, by definition, any one of a set of equally likely outcomes is exactly as likely to occur as any of the others, it mattersnot a jot how you came to select it).

Because of the reasons outlined in the article.

Now what it will effect is the distribution of winnings across the various possibilities (since the distribution is not now uniform, some sets of numbers are more likely to be selected than others, so these are more likely now to be selected by two or more people, and thus will be more likely to produce lower payouts - assuming that the amount won decreases with the number of winners - a very safe assumption with a lottery). However, I suspect that the effect on big wins will be very small (since so few people have all the numbers on any one week anyway), and that the effect was much smaller than the skew produced by human tendencies (I.e. picking birthdays, not picking groups of numbers close together etc), and of course those who pick the numbers not favoured by the machine will now be more likely to get a bigger payout, so the average winnings won't change across the whole group of consumers.

The point is not how big or small the effect will be, but that there is an effect. It totally blows the idea of an unbiased lotto out of the water.

Is there an online translator that can translate this language? The page looks like it's just showing numbers picked that were winners.

No, the top dropdown is the distribution. "Alle år" = All years.

There seams to be a significant skew at the top end but I don't know if this is a skew in how often the number 36 comes up as a winner or if the players just have some aversion to playing the number 36.

The number 35 and 36 were added later.

 

[sthomson]

Because of the reasons outlined in the article.

The point is not how big or small the effect will be, but that there is an effect. It totally blows the idea of an unbiased lotto out of the water.

You mean the article you quoted earlier? Here's what you quoted:

Since the assigned numbers "lump together", the consequence is that the winnings will be bigger, when the numbers 1-9 are overrepresented in the drawn numbers. On the other hand, the winnings will be lower, when the two-digit numbers are overrepresented, because there are proportionally more people sharing the winnings, due to the random number generator.

The gamblers hit by this have therefore combined gotten a lesser share in the average bigger winnings. On the other hand they have gotten a bigger share, combined, in the relatively lesser winnings paid out, when 1-digit numbers to a minor degree are part of the drawn numbers.

This doesn't make any sense to me. Can you explain what the article means when it says online gablers have gotten a bigger share "in the relatively lesser winnings payed out".

 

[drkitten]

This doesn't make any sense to me. Can you explain what the article means when it says online gablers have gotten a bigger share "in the relatively lesser winnings payed out".

The more money payed out in the "match-3" minor prizes, the left that is left over for current and future jackpots. So a jackpot winner today benefits from last week's smaller payout in the match-3's.

 

[Rasmus]

This doesn't make any sense to me. Can you explain what the article means when it says online gablers have gotten a bigger share "in the relatively lesser winnings payed out".

My guess:

Assume there 100 winners for the smallest price, netting them 10€ each - on average, per week.

Now, with the biased numbers being generated

- in some weeks the (fairly drawn) winning numbers will contain "many" values <10. Since those numbers are picked less often than with a normal average, there will only be 70 winners, receiving 14,29€


- in other weeks, the (fairly drawn) winning numbers will contain "few" values <10. Then, 120 players would win 8,33€.

If I had done the math correctly, the expected winnings of every player would still be the same, though.

("few" and "many", of course, depend on hopw many numbers <10 we would expect to see in an average game.)

 

[GreedyAlgorithm]

no no no no no no no.

CF why are you continuing to argue? Maths isn't like other topics where one can play with sophistry until the opposing poster grows tired of the games....

in the OP example

1)It does not affect your odds of winning.
2)It will have an effect upon your EV (though given the tendency for people to over pick 1-9, a skew which favours 10+ may actually be beneficial dependent on the stats)

that is it. Many many people have now said the same thing. It is incontrovertible.

You seem to be under the impression that I am arguing against 1 and 2. I'm not.

Now I'm confused. I think most of us thought you were arguing against 1 (the OP indicates this) and accidentally confusing it with 2.

So if you accept 1 and 2, what are you arguing for/against/whatever? And if you say something that indicates you don't have a firm grasp of what probability is, you will force me to start a new thread discussing E. T. Jaynes' take on the issue. And I think I'll do that anyway. So there.

 

[andyandy]

Some lottery analysis on EV.....

Perhaps predictably the number 7 was the most popular, chosen 25% more often than 46, the least popular. But there were surprises. Choosing numbers at the edge of the game card lifts your chance of being a solo winner, as people tend to shy away from these. Also, avoid the sequence 1, 2, 3, 4, 5, 6, Dr Simon Cox told the BBC - 10,000 people a week select this combination.



There are 13,983,816 possible combinations of the six-numbered lottery tickets
Further tips for maximising any win are to avoid numbers below 31, as these are frequently picked to match birthdays. Choosing consecutive numbers is a good ploy, as people tend to spread their lottery numbers out.

The remarkable draw on 14 November 1995 when 133 tickets shared the £16 million jackpot prize is a clear example of the effects the team had deduced.

The winning numbers were 7, 17, 23, 32, 38, 42 and 48, all of which lie in central columns of the ticket, and the players won only £120,000 each. The average number of jackpot winners is five and the average amount won is £2 million.

The information has allowed the team to show that choosing a ticket at random produces a long term winnings of 45 pence in the pound. For a very unpopular combination of numbers such as 26, 34, 44, 46, 47 and 49 the return can be raised to about 95 pence in the pound.

http://news.bbc.co.uk/1/hi/sci/tech/240734.stm

Given that I would expect Danes to be similar to Brits, I'd think that a skew away from 1-9 most likely increased EV.

 

[CFLarsen]

You mean the article you quoted earlier? Here's what you quoted:


This doesn't make any sense to me. Can you explain what the article means when it says online gablers have gotten a bigger share "in the relatively lesser winnings payed out".

...in the case where 1-9 to a minor degree are part of the drawn numbers, yes.

Did that help?

Now I'm confused. I think most of us thought you were arguing against 1 (the OP indicates this) and accidentally confusing it with 2.

Did I? In the OP, I said:

They claim that this didn't influence your chances of winning. Is that right?

And, later, I say:

Therefore, the chances of getting all 7 right must be lower, if you had chosen an automated filled-out coupon.

Right?

I don't see how you can interpret questions about X as arguing X. I am asking if certain lines of reasoning are correct.

But, applied equally to all posters, it should make for some lively - albeit fruitless - discussions.

So if you accept 1 and 2, what are you arguing for/against/whatever?

The explanation from Ingeniøren works fine for me.

And if you say something that indicates you don't have a firm grasp of what probability is, you will force me to start a new thread discussing E. T. Jaynes' take on the issue. And I think I'll do that anyway. So there.

Fine with me.

 

[Rasmus]

Something else just occurs to me:

Suppose you buy a ticket that gets automaticially filled by the malfunctioning system.

Now what?

You lose. In most cases, you simply will not win any money. What could you complain about? That you should have gotten different numbers? Possibly. But even if with a flawless system, you might have gotten the same numbers - and no matter what numbers you got, you still would have most likely lost.

You win. Now, would you complain that your price was too high or too low - proving either should be difficult to do. But assume the system did deal you a ticket with the numbers now more likely to be on a ticket, and you have to share your price with more people. The problem then is, that if the machine had been working correctly you most likely wouldn't have won. (Why should your ticket be the one that would have won if the number generator had been working as was intended?)

So, that leaves us with people who picked their own numbers, won, and had to share their price with more people than they otherwise would have. Did they get any guarantees that their numbers wouldn't be played by other players? I somehow doubt that ... but it seems those players are the ones with the best reason to complain, and quite possible those with least chance of gaining anything from it.

 

[MRC_Hans]

However you arrive at the numbers on your ticket, the chance it will win is the same.

Even if the ticket printer in the shops always printed 1234567, your chance of winning would be no different from a truly random system. If you did win, however, you would be more likely to end up sharing the prize with other people who used the same machine.

This is the basic answer.

Your chance for winning on a given ticket will be unchanged.

The distribution of wins between auto-filled tickets and hand-filled tickets will be affected. This is because the distributions of the two random generators don't match. As prizes are created by the sharing of pots, the sizes of wins will, in theory, be affected. Whether this makes any practical difference is another matter.

As for correcting any error, it is impossible, because you can never say which numbers individual players would have gotten, had the PRNG not been flawed.

Hans

 

[Cuddles]

Therefore, the chances of getting all 7 right must be lower, if you had chosen an automated filled-out coupon.

Right?

Which is wrong, and is exactly what 1 said and what everyone has been arguing against. It does not affect your chance of winning in any way, not for 7 numbers, not for 1 number.

 

[Lothian]

Which is wrong, and is exactly what 1 said and what everyone has been arguing against. It does not affect your chance of winning in any way, not for 7 numbers, not for 1 number.

I think Claus is saying that by putting ‘Right ?” after the statement he was asking a question rather than arguing for a position.

Consider the following

Teacher - What is 2 + 2 ?

Student – Is it 3 ?

Teacher - No you are wrong it is 4.

Student – No, I am not wrong I didn’t say it was three I was merely asking a question.

While Claus and the student above may be technically right in not being wrong it is fair to assume that the child thought the answer was 3 and Claus thought the biased lucky dip machine made a difference to the chances of winning. Either that or he has a really weird way of communicating.

 

[Dan O.]

I'd still like to know how one would accidently create a system for picking numbers that under represented the single digits. The simple random number generators that are skewed tend to over represent the lower numbers. Folding a larger binary range over the range of numbers to pick will also leave some numbers at the high end under represented. An extreme case in 1998, Arizona's computer for picking the winning combination for a pick 3 (from 1 through 9) never picked the number 9.