Irish lottery record 54-draw rollover, new legislation....

[Fisic]

TL;DR Summary

The Government asked to bring in legislation to make it easier to win

Our lottery here in Ireland has not been won for 27 weeks (54 draws), and it's causing consternation. Even the Government are getting involved now, but how unusual is it really?

https://www.rte.ie/news/ireland/2021/1215/1266999-ireland-lottery-latest/

Choose any 6 numbers from 47, with the odds of winning 10.7 million to one. The population of the Republic of Ireland is 5 million. The jackpot hasn't been won since early June, and has been capped at €19.1 million for some time now. The probability of no win in 54 draws is the same as rolling a die 37 times and not getting a 6.

The Lottery has been criticised for making it too difficult to win for a country of our size, but is that a reasonable argument? What would the optimum number of balls be for a population of 5 million? The extra money each draw since the cap has been filtered down to the lower prizes, but the idea being spoken of now is that these lower prizes will share the jackpot itself.

I know that each draw is an independent event and nothing is amiss, but of course the conspiracy theorists are out in force to argue otherwise.

 

[Drakkith]

Our lottery here in Ireland has not been won for 27 weeks (54 draws), and it's causing consternation. Even the Government are getting involved now, but how unusual is it really?

I don't know the math, but it sounds reasonable given your population size and number of choices for the numbers. Most of the popular lotteries here in the US have 6 number choices but our population is about 60x yours. Even here we've had plenty of no-win streaks. Of course, I think we have upwards of 60 numbers, not 47.

The jackpot hasn't been won since early June, and has been capped at €19.1 million for some time now.

Wow, I'm not sure ours has a cap. It's been over a billion USD several times: https://www.usnews.com/news/us/articles/2021-10-05/what-are-the-10-largest-us-lottery-jackpots-ever-won

 

[TeethWhitener]

Sounds like a Poisson distribution to me. If you assume that all 5 million people play the lottery every time it gets drawn, the chance of someone—possibly multiple people—winning on any given draw is (check my math):
$$1-\left(1-\frac{1}{\binom{47}{6}}\right)^{5000000}\approx 0.372$$
This is the rate of wins per draw. So the expected number of wins after 54 draws is $0.372\times54\approx20.1$. For a Poisson process, the probability of zero wins over this time period is therefore:
$$P(X=0)=e^{-20.1}\approx1.87\times10^{-9}$$
So yes it seems extremely unlikely that this is by chance. However, I have no idea if all 5 million people in Ireland play the lottery twice a week (and I suspect they don’t), so a better estimate of that number might give a more reasonable probability for zero wins. (For example, if we lower the number of players from 5 million to 1 million—possibly still too high—we get a probability of no wins after 54 draws of about 0.8%. Rare, to be sure, but not completely unheard of.)

Edit: I knew I was going to screw up the math. Hopefully it’s correct now.

 

[Drakkith]

@TeethWhitener Does your math include the possibility of there being identical sets of numbers chosen by multiple people?

 

[TeethWhitener]

@TeethWhitener Does your math include the possibility of there being identical sets of numbers chosen by multiple people?

Here’s what I did: the chance of a single person winning the lottery (assuming one person=one play) is $$\frac{1}{\binom{47}{6}}$$
(Edit: this also assumes each number appears only once and is drawn without replacement)
So the chance of that person not winning the lottery is $$1-\frac{1}{\binom{47}{6}}$$
If $n$ people play the lottery, the chance of none of them winning is $$\left(1-\frac{1}{\binom{47}{6}}\right)^n$$
So the chance of at least one person winning the lottery (could be multiple winners) is $$1-\left(1-\frac{1}{\binom{47}{6}}\right)^n$$
Of course, it may be the case that a person buys more than one ticket, in which case the math above merely describes the probability that the lottery is won given $n$ tickets sold.

 

[Fisic]

Here’s what I did: the chance of a single person winning the lottery (assuming one person=one play) is $$\frac{1}{\binom{47}{6}}$$
(Edit: this also assumes each number appears only once and is drawn without replacement)
So the chance of that person not winning the lottery is $$1-\frac{1}{\binom{47}{6}}$$
If $n$ people play the lottery, the chance of none of them winning is $$\left(1-\frac{1}{\binom{47}{6}}\right)^n$$
So the chance of at least one person winning the lottery (could be multiple winners) is $$1-\left(1-\frac{1}{\binom{47}{6}}\right)^n$$
Of course, it may be the case that a person buys more than one ticket, in which case the math above merely describes the probability that the lottery is won given $n$ tickets sold.

I should point out that the minimum play per person is two lines and around 2.2 million people play each draw (but probably more in recent weeks). That makes the lack of wins all the more surprising.

 

[sysprog]

I think that 'capping' the prize is unvarnished unmitigated theft. If they want a €19.1 million single-prize ceiling, then they should allocate the amount in excess of that to the next prize(s). For them to thievingly arrogate the 'excess' into their coffer, is a gross violation of the public trust, and of their contract with each and every ticket purchaser.

 

[Fisic]

They do indeed allocate the extra takings to the lower prizes. The cap occurs when the jackpot exceeds the previous high record (€18.6M).

We're now into 2022 and it still hasn't been won. One of these days/weeks/months...

 

[BWV]

I think that 'capping' the prize is unvarnished unmitigated theft. If they want a €19.1 million single-prize ceiling, then they should allocate the amount in excess of that to the next prize(s). For them to thievingly arrogate the 'excess' into their coffer, is a gross violation of the public trust, and of their contract with each and every ticket purchaser.

They don't call it an idiot tax for nothing

 

[BWV]

Sounds like a Poisson distribution to me. If you assume that all 5 million people play the lottery every time it gets drawn, the chance of someone—possibly multiple people—winning on any given draw is (check my math):
$$1-\left(1-\frac{1}{\binom{47}{6}}\right)^{5000000}\approx 0.372$$
This is the rate of wins per draw. So the expected number of wins after 54 draws is $0.372\times54\approx20.1$. For a Poisson process, the probability of zero wins over this time period is therefore:
$$P(X=0)=e^{-20.1}\approx1.87\times10^{-9}$$
So yes it seems extremely unlikely that this is by chance. However, I have no idea if all 5 million people in Ireland play the lottery twice a week (and I suspect they don’t), so a better estimate of that number might give a more reasonable probability for zero wins. (For example, if we lower the number of players from 5 million to 1 million—possibly still too high—we get a probability of no wins after 54 draws of about 0.8%. Rare, to be sure, but not completely unheard of.)

Edit: I knew I was going to screw up the math. Hopefully it’s correct now.

Why a Poisson? wouldn't the odds of no one winning after 54 draws be (1-.37)^54 (which is an even smaller number)?

But it does not look like they sell that many tickets, the article says the rollover odds on a draw last month were 87% with 1.4M tickets sold

https://www.irishtimes.com/news/ire...otto-draw-a-case-of-48th-time-lucky-1.4743186

At 87% rollover odds 54 rollovers is a 0.05% chance, small but not a lifetime-of-the universe type number

 

[TeethWhitener]

Why a Poisson?

1) The events are discrete.
2) The drawings are (presumably) independent from one another.
3) The probability of winning is constant over time (equivalent to a constant expected rate of winning).
Those conditions pretty much define a Poisson distribution.

 

[BWV]

1) The events are discrete.
2) The drawings are (presumably) independent from one another.
3) The probability of winning is constant over time (equivalent to a constant expected rate of winning).
Those conditions pretty much define a Poisson distribution.

But the Poisson is in continuous time, this is a binominal distribution - analogous to a coin-flipping exercise

​

 

[TeethWhitener]

But the Poisson is in continuous time, this is a binominal distribution - analogous to a coin-flipping exercise

​

Good point. How much difference does it end up making?

 

[BWV]

Good point. How much difference does it end up making?

the difference between 1.87 *10^-9 and (1-.37)^54, which is 1.5*10^-11, which is either a factor of 100 or completely inconsequential given the low probability

 

[Melbourne Guy]

I think that 'capping' the prize is unvarnished unmitigated theft.

Why so, @sysprog?

Like many state-run lotteries, Ireland's National Lottery was established in 1986 to raise funds for good causes, so assuming the funds do go to good causes (you can judge for yourself here: https://www.lottery.ie/news/good-causes) then its purpose is a social good.

Even without that, there is no 'theft'. Punters know what they are getting into, they are not forced to play, and there is no scam going on. Indeed, the cap is hard to miss with even a cursory surf of their site!

 

[sysprog]

Why so, @sysprog?

Like many state-run lotteries, Ireland's National Lottery was established in 1986 to raise funds for good causes, so assuming the funds do go to good causes (you can judge for yourself here: https://www.lottery.ie/news/good-causes) then its purpose is a social good.

Even without that, there is no 'theft'. Punters know what they are getting into, they are not forced to play, and there is no scam going on. Indeed, the cap is hard to miss with even a cursory surf of their site!

The ceiling wasn't part of the rules up front, wherefore, imposing it after the ticket purchases that were made under the existing rules is theft. If I buy a lottery ticket, and nobody wins, then part of my payment goes to sweeten the pot for the next drawing. Taking the 'excess' by subsequent imposition of a ceiling is plainly theft.

 

[Melbourne Guy]

The ceiling wasn't part of the rules up front

When did they impose the cap?

 

[sysprog]

When did they impose the cap?

Not between 7:45pm and 7:55pm, immediately after a win, while tickets were not being sold, which would have been the only legitimate time to implement such a rule change.

 

[Office_Shredder]

The ceiling wasn't part of the rules up front, wherefore, imposing it after the ticket purchases that were made under the existing rules is theft. If I buy a lottery ticket, and nobody wins, then part of my payment goes to sweeten the pot for the next drawing. Taking the 'excess' by subsequent imposition of a ceiling is plainly theft.

Take a chill pill, you are way too worked up over this capping.

1.) I'm still confused how you know e.g this rule was not introduced as part of when the lottery was first created.

2.)
https://www.lottery.ie/how-to-play/lotto/new-lotto-cap

When the jackpot reaches a cap, it is fixed and guaranteed at that level. This means that any excess amount above the Jackpot that would normally increase the jackpot size does not, but flows down to the next tier at which there are prize winners.

I don't really understand *why* they do this, but apparently they do.

 

[sysprog]

When did they impose the cap?

Not between 7:45pm and 7:55pm, immediately after a win, while tickets were not being sold, which would have been the only legitimate time to implement such a rule change.

Take a chill pill, you are way too worked up over this capping.

1.) I'm still confused how you know e.g this rule was not introduced as part of when the lottery was first created.

2.)
https://www.lottery.ie/how-to-play/lotto/new-lotto-cap
I don't really understand *why* they do this, but apparently they do.

I feel like Emily Lytella $-$ I didn't know about the flowing down to the next levels $-$ consequently I abjure my denunciation $-$ "never mind" . . .

 

[Melbourne Guy]

I had to look up Emily Lytella, @sysprog, still not sure what the reference means, but I agree with @Office_Shredder in that it's not obvious that the rules were changed. It seems that a cap has always been available to the lottery, it's just not necessary most of the time. But as you've abjured your denunciation, it's all good and hopefully someone wins the damned thing at the next draw and the conspiracy theorists can settle and focus on more important issues. After all, January 6 is coming up fast

 

[sysprog]

I had to look up Emily Lytella, @sysprog, still not sure what the reference means, but I agree with @Office_Shredder in that it's not obvious that the rules were changed. It seems that a cap has always been available to the lottery, it's just not necessary most of the time. But as you've abjured your denunciation, it's all good and hopefully someone wins the damned thing at the next draw and the conspiracy theorists can settle and focus on more important issues. After all, January 6 is coming up fast

Emily Lytella is a character, played by Gilda Radner, from the early days of 'Saturday Night Live' $-$ her routine is that she miinterprets something comparatively innocuous as something pernicious, and starts into an indignant diatribe, only to be told by a colleage that she has mistaken one word for another, whereupon she apologizes and says "never mind". Mine was a different kind of getting it wrong, but similar in some other respects $-$ here's an example of Emily Lytella $-$

 

[Fisic]

Well, wouldn't you know, this lottery was finally won on Saturday...right on the day that they made it a must-win draw.

Since June, nobody had matched the 6 numbers drawn on any of the twice-weekly draws, but the rules for last Saturday's draw meant that it would no longer be rolled over. If 6 numbers were not matched, the jackpot would instead be shared out among the Match-5 or Match-4 + Bonus Number winners. And of course, as if by magic, one ticket in Co. Mayo did manage to match 6 and hence pocket the €19.06M sum. What are the chances...literally?

I'm not saying anything has been underhanded throughout all this process, but it would be interesting to see what is the probability of it turning out this way. All draws are independent and carry the same win-probability, yet what are the chances that the first match-6 ticket in 6 months and 63 draws would occur right on the day it was going to be split among the lower winning tickets?

 

[PeroK]

Well, wouldn't you know, this lottery was finally won on Saturday...right on the day that they made it a must-win draw.

Since June, nobody had matched the 6 numbers drawn on any of the twice-weekly draws, but the rules for last Saturday's draw meant that it would no longer be rolled over. If 6 numbers were not matched, the jackpot would instead be shared out among the Match-5 or Match-4 + Bonus Number winners. And of course, as if by magic, one ticket in Co. Mayo did manage to match 6 and hence pocket the €19.06M sum. What are the chances...literally?

The key piece of data is how often the lottery is won? I can't immediately find the figures, but assume it's normally won one draw in ten. That means the probability of a rollover is $0.9$.

How likely is it that it is won on the last chance? I guess that means about 60 draws without a jackpot and then one draw with a jackpot. Using $0.9$ as above the probability of a last chance win would be approximately $0.018 \%$. That's about one chance in 5,500.

When you ask how likely that is, then the first answer is that it's inevitable! Inevitable that it will happen eventually if the lottery takes place for long enough.

The next thing to calculate, therefore, is how often that would occur. Again using the figure $0.9$, the average run before a jackpot would be 10 draws. You can get this by analysing the binomial distribution. That's about 5 weeks. Does this makes sense? If not, we need to adjust that figure of $0.9$.

In this case, the last chance jackpot should occur every 55,000 draws or 27,500 weeks. That's about every 500 years.

What you've witnessed would appear to be something that only happens every 500 years on average.

But, there are hundreds of lotteries around the world. Let's say there are 50 in Europe. So, we might expect something this to happen somewhere in Europe every 10 years or so.

If we record all lotteries around Europe then these rare events will happen from time to time. That brings us back to the question of what evidence there is that this rare event in Ireland was staged? Otherwise, it's just one of those rare events, one of which is bound to happen somewhere every few years.

 

[Fisic]

In the end there were 63 consecutive draws without a win. Back at the start of December, after 47 of these draws, a statistician made the following statement (from Wikipedia)

Interviewed on Newstalk morning radio, Michael Cronin, head of statistics at the School of Mathematical Sciences at University College Cork, stated that calls for an investigation were unwarranted, noting that the odds of 47 consecutive Lotto rollovers were about 1,500 to one, roughly comparable to the odds of being born on the 29th of February or getting four consecutive sixes when throwing a dice.[43]

Of course, consecutive winning draws are also statistically unlikely.

Regardless, on 14 December 2021, National Lottery regulator Carol Boate and representatives of Premier Lotteries Ireland appeared before the Oireachtas Finance Committee to address the situation.[44] Boate reassured committee members that there had been nothing amiss with the Lotto game since June, stating that "statistically unlikely events are part of the nature of games of chance and lotteries", and observing that similar extended sequences of rollovers had happened in other countries' lotteries.[45] She noted that before the lengthy series of rollovers, the Lotto jackpot had been won for three Saturdays in a row, on 22 May, 29 May, and 5 June 2021, which was also a statistically unlikely occurrence.[46] PLI's chief executive Andrew Algeo informed the committee that PLI was seeking regulatory approval for a "must-win" draw in scenarios where a capped jackpot had not been won for an extended period.[47]

New rules from here on state that there will now be a limit of 5 winless draws from when a jackpot record is reached.

 

[PeroK]

Note that if the number of tickets bought every week changes significantly, then we would need that data. So, assuming that approximately the same number of tickets is bought every week, we can update the figures.

If the odds of 47 rollovers is 1/1500, then that gives the probability of a rollover of only $0.856$. The probability 63 consecutive rollovers followed by a jackpot is only about 1/125000. This equates to an event that happens only every 8,000 years(!).

Now, of course, there are other things that could have happened: like a run of consecutive jackpots or, more simply, a run of 64 draws without a win. This is not the only rare event that would have been noticed. And, it's hard to estimate how many such rare events there are! Nevertheless, I think we are looking an extraordinary turn of events over the past few months.

Again, in order to make anything of it, you need a plausible hypothesis of what might have caused it.