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# Probability of winning the irish lottery

• 22-10-2010 11:04am
Closed Accounts Posts: 353 ✭✭

Someone in my office is setting up a syndicate for the Irish lottery, she said the jackpot is 16 million euros.

What's the probability of winning the lottery ?

For an investment of 5 euros in tickets (the amount we are required to invest), what is the expected value of the return ?

Is it more or less than 5 euros

I know it depends on the probability of winning and the total number of tickets bought........but can someone do the maths for me !

I'll join the syndicate if it makes mathematical sense.

• Closed Accounts Posts: 6,081 ✭✭✭

The odds of winning the Irish lotto are 1 in 8,145,060. i.e 1 divided by 45 choose 6. But I know nothing about how syndicates work or anything like that as I've played the lotto once in my life so maybe another poster can help you

• Closed Accounts Posts: 39,022 ✭✭✭✭

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• Registered Users Posts: 68,318 ✭✭✭✭

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.

In the syndicate, if there are nine of you paying a fiver each, then you can buy 30 tickets. This improves the chances of winning to a paltry 1 in 271,502.

I'm not sure what you mean by "expected return". You should expect to get zero. The "possible return" if there are nine in the syndicate is 350,000%.

Interestingly, the odds of winning "something" are about 1 in 40. In your syndicate of nine, this ups the odd significantly of winning "something" to about 75%. However this "something" is a small amount - usually about a fiver, divided by nine people, represents a return of 11% of your original sum. Depending on what way you want to look at it, your likely ROI is about -95%, that is, you stand to lose about 95% of your original investment, though you may lose all of it.

• Closed Accounts Posts: 39,022 ✭✭✭✭

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• Registered Users Posts: 2,793 ✭✭✭

This is based on the UK lotto, but an interesting little tool.

http://lottery.reevo.com/

• Closed Accounts Posts: 6,943 ✭✭✭

Token post:

But someone has to win!!!!

• Registered Users Posts: 68,318 ✭✭✭✭

Pfft, let's not complicate it

I should add that syndicates aren't an entirely bad idea. In the example I give above, playing 3 lines on your own your chances of winning €16 million all to yourself is roughly 1 in 2.7 million. On the other hand, 10 people playing 30 lines has ten times better odds of winning, but the amount you win is only €1.6m each. Obviously here the smart choice is the same amount of investment, with a better chance of return, given the levels of money involved.

• Registered Users Posts: 10,135 ✭✭✭✭

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

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• Registered Users Posts: 399 ✭✭

Interestingly the odds of dying (on average) any hour in Ireland are 1 in 1.1 million (Death rate 4 per hour). So therefore If you do the lottery more than ~ 8 hours before the draw you have a higher chance of being dead before the draw than winning.

• Closed Accounts Posts: 353 ✭✭

Thanks everyone , 8 million to 1 against..........

If you bought every combination of numbers, it would cost 12 million

The jackpost is 16 million, therefore buying a ticket tonight could make mathematical sense

The expected return (probably the wrong term) on a 1.50 investment is 2 euros (that would be true if you were the only person who bought a ticket)........it's actually slightly higher than 2 euros because of the match 5s and match 4s.

But of course there is a risk that someone else might win too, and the jackpot might be divided among several people

I think buying a ticket for a 16 million jackpot probably makes sense, but it depends on how many other tickets are competing for the jackpot.

• Closed Accounts Posts: 353 ✭✭

Dermot2468 wrote: »
Interestingly the odds of dying (on average) any hour in Ireland are 1 in 1.1 million (Death rate 4 per hour). So therefore If you do the lottery more than ~ 8 hours before the draw you have a higher chance of being dead before the draw than winning.

That's definitely true if you are aged 80, but if you are a 17 year old who doesn't drive or who isn't on a rock climbing holiday, I think the odds of winning the lottery are higher than being dead in the 8 hours before the lottery.

• Closed Accounts Posts: 39,022 ✭✭✭✭

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• Closed Accounts Posts: 353 ✭✭

I think I have the maths

12 million is the figure you'd want to win for overcoming odds of 8'000'000 to one if you buy 1 ticket. (for it to be a neutral bet)

But if you do overcome the odds of 8'000'000 to 1, can you expect to win 12'000'000 ?

Because there is a probability that someone else will also win....

The probability of someone else winnng is

number of tickets sold / 8'000'000

Let's say there are 3'000'000 tickets sold, there is a 36% chance that if you win, you'd have to share it with someone else

Therefore the prize for overcoming the 8'000'000 to 1 odds is reduced from 16'000'000 to say (16'000'000 minus 0.36*16'000'000) = 10.3 million

If you overcome odds of 8 million to 1, you can expect to get 10.3 million in return, when you'd need to get 12 million to 1 for it to be a fair value bet...

I'm not joining the syndicate ! I'm sticking to poker

• Registered Users Posts: 12,821 ✭✭✭✭

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.

We are afraid to put men to live and trade each on his own private stock of reason; because we suspect that this stock in each man is small.

-- Edmund Burke

• Registered Users Posts: 36,128 ✭✭✭✭

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.

• Registered Users Posts: 5,143 ✭✭✭

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

• Moderators, Science, Health & Environment Moderators Posts: 1,835 Mod ✭✭✭✭

Yakuza wrote: »
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

Yeah, and they added several new numbers - thus vastly increasing the number of possible combinations and making any brute force attempts effectively impossible.

They marketed this on TV as:

"you know the way you couldn't pick numbers 37, 38 and 39 - well the nice people at the National Lottery have now added these for your enjoyment..."

and the majority of the public went

"Ah sure that's great!"

• Registered Users Posts: 4,586 ✭✭✭

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?

• Registered Users Posts: 1,015 ✭✭✭

This post has been deleted.

They should have you as guest speaker for all the lotto winners bash we've had through the years!;)

• Closed Accounts Posts: 834 ✭✭✭

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?

???

Population ~ 4,500,000

$\frac{1}{8145060} \times 4500000 = 55.25$%

• Moderators, Science, Health & Environment Moderators Posts: 1,835 Mod ✭✭✭✭

Reillyman wrote: »
???

Population ~ 4,500,000

$\frac{1}{8145060} \times 4500000 = 55.25$%

Maybe sock puppet's refering to the fact that you have to buy at least two lines at a time?

• Registered Users Posts: 4,586 ✭✭✭

Reillyman wrote: »
???

Population ~ 4,500,000

$\frac{1}{8145060} \times 4500000 = 55.25$%

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?

• Moderators, Science, Health & Environment Moderators Posts: 1,835 Mod ✭✭✭✭

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?

Nope, the probability of at least one person winning is maximized if we all pick different numbers. Since this doesn't happen in actuality, the probability of at least one person winning the jackpot is less than

$N_{players} \cdot \frac{1}{8145060}$

This assumes that the numbers are random and there's no fiddling going on of course

• Registered Users Posts: 2,122 ✭✭✭

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.

• Closed Accounts Posts: 23,316 ✭✭✭✭

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

• Registered Users Posts: 2,122 ✭✭✭

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

• Registered Users Posts: 4,586 ✭✭✭

Nope, the probability of at least one person winning is maximized if we all pick different numbers. Since this doesn't happen in actuality, the probability of at least one person winning the jackpot is less than

$N_{players} \cdot \frac{1}{8145060}$

This assumes that the numbers are random and there's no fiddling going on of course

Yeah I see how I went wrong now. I was thinking of the probability of any 2 players picking the same number. Pretty big mistake:pac:

• Registered Users Posts: 332 ✭✭

i never thought there were such odds against you winning the lotto!! :eek:

• Registered Users Posts: 68,318 ✭✭✭✭

Yeah I see how I went wrong now. I was thinking of the probability of any 2 players picking the same number. Pretty big mistake:pac:
Funnily enough, the nature of humans is such that the odds of two people picking the same six numbers are shockingly high.
We tend to have an inbuilt penchant for patterns, so if you pick a common sequence like 1-6 or 2,4,6,8,10,12, or anything remotely similar you are guaranteed to only win a very tiny amount of the jackpot as a surprisingly large amount of people pick these sequences. Presumably because they think that nobody else is clever enough to do that.

I'm almost sure I've read that this happened somewhere - the numbers 1 - 6 came out in the lottery draw and about 50 people turned up, quite disappointed with their share of the jackpot

If you want to ensure that you have the best chance of being the sole winner of a jackpot, use a quickpick. The odds of your quickpick matching someone else's ticket are exactly the same as the odds of winning the lotto.