Probability of winning the irish lottery

Those are just the odds of winning the top prize. It's much, much more complicated than that, especially when you take in the Lotto Plus draws (€2 a line). Here are the results of last Wednesday's draws, with the number of winners and prize values:
Winners|Numbers Matched|Prize
Lotto
0|Jackpot Winner|€14,764,690
4|Match 5 + Bonus|€25,000
86|Match 5||€1,761
225|Match 4 + Bonus|€168
4,433|Match 4|€53
5,196|Match 3 + Bonus|€30
71,881|Match 3|€5 scratchcard
LottoPlus2
0|Top Prize Winner|€350,000
1|Match 5 + Bonus|€3,500
93|Match 5|€350
180|Match 4 + Bonus|€35
3,772|Match 4|€15
5,341|Match 3 + Bonus|€9
62,453|Match 3|€3 scratchcard
LottoPlus2
1|Top Prize Winner|€250,000
2|Match 5 + Bonus|€2,500
75|Match 5|€250
209|Match 4 + Bonus|€25
3,716|Match 4|€10
5,215|Match 3 + Bonus|€5
60,388|Match 3|€1 scratchcard

Plus, of course, the fact that no Lottery pays out all its winnings. Not even close: they're raising money for various causes in the process. From the National Lottery website:

The National Lottery's mission is to operate a world class lottery raising funds for good causes on behalf of the Government and we are proud to have delivered on this again in 2009, raising €263.5 million.
This brings the total amount raised since the establishment of the National Lottery to over €3.4 billion

That's €3.4 billion that went to all kinds of things that would otherwise be funded by taxation, and didn't go to lottery winners. In other words: lotteries are a tax on those who don't know basic statistics.

Lethal_Bullet wrote: »

Remember with the buy every combination thing, it cant actually be done. AFAIK people have tried in the past but the NL reserve the right to shut terminals at their will.

The logistics would be horrendous - I can't remember if there's 8 or 10 boxes on a slip; you'd need to either have (roughly) 1,000,000 or 800,000 slips filled out (this could be done with a printer and software).
The longest gap between draws is roughly 3 days 23 hours 40 minutes (iirc the terminals close at 7:45 so you would have approx from 8:05 on the Sat night to 7:45 on the Weds night). That's 344400 seconds. At a conservative estimate of 5 seconds per slip (to insert the slip into the reader, read, print the lotto ticket and eject the slip), that gives you 68880 print cycles (assuming you can find a 24-hour shop *and* the shopkeeper is willing to work for almost 4 days straight ). To print off all 1m (800k) slips would need 15 (12) people working continuously for 4 days. There is no way that would go unnoticed by the either the general public or the NL themselves. As being somewhat knowledgeable in the field of IT, I've no doubt the NL would have algorithms to spot unusual buying patterns coming up to a draw.

To top it all off, if the top prize was shared, you'd be buggered. The pools for the match 5 etc is proportional to the main prize fund, so you'd just force down the individual prize amount for those ones. The only way to make money would be to play would have been when there was a (relatively large) fixed prize for the lesser results (match 4 etc). When the Scruffy Murphy syndicate attempted this in the early 90's, it was around a bank holiday when there was £100 for the match 4 (iirc). Shortly after this attempt, the draw went bi-weekly

seamus wrote: »

The odd are 1 in 8,145,060.

This means that if everyone in Ireland bought a single ticket tonight with different numbers on them, there is still roughly a 50% chance that no-one will win.

That's a very unrealistic assumption though. In a real-life scenario would you not be practically guarenteed a winner if that number of people played?

Reillyman wrote: »

???

Population ~ 4,500,000

[latex]\frac{1}{8145060} \times 4500000 = 55.25[/latex]%

But that has everyone picking different numbers though which is why I asked the question. If it was a real-life scenario where it's possible for more than person to pick each combination would the probability of at least one person winning the jackpot not be very high?


edit: I'm not sure whether I'm thinking about it right or not. Is the logic the same as that for the birthday problem or do I have it wrong?

sock puppet wrote: »

But that has everyone picking different numbers though which is why I asked the question. If it was a real-life scenario where it's possible for more than person to pick each combination would the probability of at least one person winning the jackpot not be very high?


edit: I'm not sure whether I'm thinking about it right or not. Is the logic the same as that for the birthday problem or do I have it wrong?

The opposite would be true. If people pick the same numbers thats less individual combinations resulting in a lower chance of the lottery being won.

In fact i know 2 people who when they do the lotto pick the numbers 1,2,3,4,5,6. They reckon they have as good a chance as any of coming up:)
They dont know each other so i never mentioned it to them.

amacachi wrote: »

The chances of winning the Lotto are just like everything else, 50/50, you either win it or you don't. :pac:

In that case ill bet you 5 euro every draw that you wont win. Ill even give ya odds of 100/1:p:p:p

seamus wrote: »

Funnily enough, the nature of humans is such that the odds of two people picking the same six numbers are shockingly high.

This is also why humans are terrible random number generators. Commonly if asked to give a sequence of random numbers we tend to choose numbers that are not round numbers, to seem more random. Of course a uniformly distributed sequence of random numbers, given enough time, will have as many 30s as 37s.

Not only that, but we tend to purposely "over jump" to seem more random, like "3, 81, 21, ...", our average distance between two numbers is too large. Also we tend to alternate between high and low, again to seem more random, however this means that the correlation between the nth number and the (n+2)th is too high.

Of course even computers fail to give random sequences if you watch them for long enough. The only real random number generators are atomic decays, e.t.c.

EDIT: The actual reason this is related to choosing a limited number of options in the Lotto is because we have a terrible natural affinity for statistics of any kind. Maths education studies show that statistics takes longer to have a concrete conceptualisation than other areas of mathematics.

acurno wrote: »

I've actually had a running debate with my Dad about the picking of numbers only from 1-31 taking into account the birthdays. He stopped doing his numbers that he did for 20 years after listening to some Mathematical punter on Ray Darcy a couple of years ago. If those numbers ever come up......

Anyway, I'm still failing to grasp the logic of it, I'm not from a mathematical background so I can claim ignorance.

Surely the probability of picking six numbers to a max of 31, are the same as picking six numbers up to 42. There's as much a chance, as has been said, of 1,2,3,4,5,6 coming out as there are any other six numbers, irregardless of whether there's 36 or 42 numbers in the draw. My Dad argues that if you leave out the numbers from 31-42, that you're automatically excluding the probability of numbers 31-42 coming up. I say that the fact that you always pick 6 numbers automatically excludes you from the probability of the remaining 36 numbers coming up.

If I don't make any sense above I'll give an example...

Line 1: 3, 7, 9, 17, 21, 25
Line 2: 2, 9, 17, 21, 25, 36

Line 1 excludes numbers above 25, Line 2 doesn't. Is my Dad right in saying Line 2 has a greater probability and statistical chance of winning than Line 1?

I say no. They have the same.

You are right, they have the same probablility of winning.

Generally, the simplest way of explaining such a situation to a lay-person is to reduce it to its simplest terms. Compare your Dad's choice to 1,2,3,4,5,6. According to your Dad, that automatically excludes the probablility of 7-45, so you are guaranteed to win. Obviously that's not true

The only effect it can have is if you do pick the right numbers based on dates etc, you are more likely to end up sharing a jackpot

Lethal_Bullet wrote: »

Remember with the buy every combination thing, it cant actually be done. AFAIK people have tried in the past but the NL reserve the right to shut terminals at their will.

It was done successfully in the early days of the lotto in Ireland back when it was a weekly draw with only 36 numbers.

In a 6/36 lottery, the odds of matching all six numbers and winning the jackpot are 1 in 1,947,792. At Lotto's initial cost of £0.50 per line, all possible combinations could be purchased for £973,896. This left Lotto vulnerable to a brute force attack, which happened when the jackpot reached £1.7 million for the May 1992 bank holiday drawing. A 28-member Dublin-based syndicate, organized and headed by Polish-Irish businessman Stefan Klincewicz, had spent six months preparing by marking combinations on almost a quarter of a million paper playslips. In the days before the drawing they tried to buy up all possible combinations and thus win all possible prizes, including the jackpot.
The National Lottery tried to foil the plan by limiting the number of tickets any single machine could sell, and by turning off the terminals Klincewicz's syndicate was known to be using heavily. Despite its efforts, the syndicate did manage to buy over 1.6 million combinations, spending an estimated £820,000 on tickets. It had the winning numbers on the night—but two other winning tickets were sold, too, so the syndicate could claim only one-third of the jackpot, or £568,682. Match-5 and match-4 prizes brought the syndicate's total winnings to approximately £1,166,000, representing a profit of approximately £310,000 before expenses.
Klincewicz later appeared on the television talk show Kenny Live and capitalized on his short-lived notoriety with a self-published lottery-system book entitled Win the Lotto.

http://en.wikipedia.org/wiki/National_Lottery_(Ireland)#Lotto_6.2F36:_1988.E2.80.9392

Deleted User wrote: »

how many numbers would you need to pick to have (exactly or approximately) a 50-50 chance of naming all six numbers drawn. Also, how many numbers would you need to pick to have a 50-50 chance of naming none of the six numbers drawn?

In the English lotto you pick 6 numbers from 49.

If you pick 5 numbers then there is a 50.5% chance that you have none of the winning numbers.

If you pick 44 numbers you have a 50.5% chance of having all the winning numbers.

(Notice that 44 = 49 - 5. Try to figure out what's going on!)




In the Irish lotto you pick 6 numbers from 45.

If you pick 5 numbers then there is a 47.1% chance that you have none of the winning numbers.

If you pick 40 numbers you have a 47.1% chance of having all the winning numbers.

(Notice that 40 = 45 - 5)



Explanation for 50.5%:

If you pick 1 number there is a (43/49) chance that it is not a winning number.

If you pick 2 numbers there is a (43/49) chance the first is not winning. If the first is not a winning number, there is a (42/48) chance the second is also not winning. So the probability both are not winning is (43/49)*(42/48).

For 5 numbers,

Answer = P1*P2*P3*P4*P5
=(43/49)*(42/48)*(41/47)*(40/46)*(39/45)
=0.505
=50.5%

where
P1 = Probability 1st number is not winning,
P2= Probability 2nd number is not winning if 1st is not winning,
P3= Probability 3rd number is not winning if 1st and 2nd are not winning,
P4= Probability 4th number is not winning if 1st, 2nd and 3rd are not winning,
P5= Probability 5th number is not winning if 1st, 2nd, 3rd and 4th are not winning.

Related:
Find out about the Birthday Problem.

A lottery in the US has broken the record for the largest lottery jackpot. Draw is in two days and it has a nominal value of almost half a billion dollars with the odds of winning the jackpot at roughly 175,000,000:1.