Settle an argument

Iwannahurl

This is a thread on After Hours.


What are the odds of winning the lotto twice?

tommy21 wrote:

Just heard on newstalk that some lucky dub had picked up a cool half mil after winning 2.5mill a few years ago. I can't help but feel a little bitter...

What are the odds of this?!

I think posters are framing the question incorrectly and are therefore coming up with a range of inaccurate (or irrelevant) answers.

IMO the question is "what are the chances that some lucky person will win the Lotto twice within a time-frame of several years" and the answer is "quite likely, actually".

I'm far from being a mathematician or Stats boffin, though.

Any thoughts? Or better still, rational analysis?

trad

It's about the same as lightening striking the same place twice.

Fremen

Well, you need to phrase the question a bit more precisely than that.

If I ask "what is the probability that *I* will win the lotto twice in a row", it's easy enough to show that it's the square of the probability of winning it once.
(the odds of me rolling a die and getting 6 are 1/6. The odds of that happening twice in a row are 1/36).

If you ask "what is the probability that someone in Ireland will win the lotto twice over ten years", the calculation is not nearly as straightforward, and you need to make some assumptions to simplify the calculation.

As a ballpark figure, suppose there are a thousand jackpot winners who still buy two lines a week. In any given week, the odds that one of them will win are about one in three thousand (assuming the lotto is still 42 numbers choose 6 like when I was a kid).

Under these slightly flaky assumptions, after about 39 years, it's 50:50 for a jackpot winner to win twice

( to see this, solve (2999/3000)^x = 1/2. Turns out x = 39 or thereabouts)

bnt

Just remember that luck has no memory. Say you won the lottery last week: what are the odds of you winning it this week? Assuming you play the same way, they are the same as they were last week, and the week before, and next week, and so on. Winning or losing does not change the odds.

Fremen

trad wrote: »

It's about the same as lightening striking the same place twice.

That's exceedingly likely. It's one of the reasons tall buildings have lightning conductors on them.

Iwannahurl

Fremen wrote: »

Well, you need to phrase the question a bit more precisely than that.

If I ask "what is the probability that *I* will win the lotto twice in a row", it's easy enough to show that it's the square of the probability of winning it once.
(the odds of me rolling a die and getting 6 are 1/6. The odds of that happening twice in a row are 1/36).

If you ask "what is the probability that someone in Ireland will win the lotto twice over ten years", the calculation is not nearly as straightforward, and you need to make some assumptions to simplify the calculation.

As a ballpark figure, suppose there are a thousand jackpot winners who still buy two lines a week. In any given week, the odds that one of them will win are about one in three thousand (assuming the lotto is still 42 numbers choose 6 like when I was a kid).

Under these slightly flaky assumptions, after about 39 years, it's 50:50 for a jackpot winner to win twice

( to see this, solve (2999/3000)^x = 1/2. Turns out x = 39 or thereabouts)

Many thanks. This is the 'real' answer, IMO.

To estimate probabilities correctly you need to determine the conditions precisely. The OP's question in the other thread was not framed in such a precise manner, hence the ambiguity as to whether he was talking about the prospective likelihood that I or any other specific person will win the lottery with two particular tickets on two particular occasions.

However, when you add to the equation some real-world assumptions that are imprecise but more than likely true -- as you have done, eg estimated number of former jackpot winners buying so many lines per draw so many times per week over so many years -- then you find that statistically it's not very surprising to see "some lucky Dub" (to quote the OP in the other thread) winning twice in the space of a few years.

For the record, it could not have been me.

hivizman

Fremen wrote: »

Well, you need to phrase the question a bit more precisely than that.

If I ask "what is the probability that *I* will win the lotto twice in a row", it's easy enough to show that it's the square of the probability of winning it once.
(the odds of me rolling a die and getting 6 are 1/6. The odds of that happening twice in a row are 1/36).

If you ask "what is the probability that someone in Ireland will win the lotto twice over ten years", the calculation is not nearly as straightforward, and you need to make some assumptions to simplify the calculation.

As a ballpark figure, suppose there are a thousand jackpot winners who still buy two lines a week. In any given week, the odds that one of them will win are about one in three thousand (assuming the lotto is still 42 numbers choose 6 like when I was a kid).

Under these slightly flaky assumptions, after about 39 years, it's 50:50 for a jackpot winner to win twice

( to see this, solve (2999/3000)^x = 1/2. Turns out x = 39 or thereabouts)

Isn't this basically a Poisson distribution?

Suppose that there are two lotteries each week for a period of 20 years - so a total of 2080 lotteries. Each lottery is independent of the others. Suppose also that, on average across the period, there is one jackpot winner per week (this averages out the weeks when there is no winner and the weeks when the jackpot is shared). Then there will be 2080 jackpot winners in the period.

Next, suppose that, on average, the same 2 million people enter the lottery each week (clearly this is a big assumption, especially over a 20 year period, but if there are some different people in the lottery each week, the probability of a multiple winner is going to be reduced).

This means that the proportion of lottery winners to the population is 2080/2000000 = 0.00104. Crudely speaking, this would mean that, over a 20 year period, anyone who enters the lottery every time would have about a 1 in 1000 chance of winning the lottery. However, this does not allow for the possibility that someone could win the lottery more than once. This is where the Poisson distribution comes in.

Calculating the probabilities for various outcomes, you get a probability of 0.998961 that a given person would never win the jackpot, a probability of 0.001038 that a given person will win the jackpot precisely once, and a probability of 0.00000054 that a given person will win the jackpot precisely twice. There is a non-zero probability that a given person will win the jackpot three or more times, but this is of the order of 0.0000000001.

In a population of 2 million, the Poisson distribution predicts that 1,997,921 people will never win the jackpot, 2,078 people will win the jackpot precisely once, and one lucky person will win the jackpot twice over a period of 20 years.

Iwannahurl

Here's a paper that addresses the Lottery Double Winner question. The relevant section, Example 2.23, is on pages 18-19 of the PDF file, 42-43 in the original document pagination.

hivizman

Iwannahurl wrote: »

Here's a paper that addresses the Lottery Double Winner question. The relevant section, Example 2.23, is on pages 18-19 of the PDF file, 42-43 in the original document pagination.

Thanks - I note that the paper treats the problem as involving a Poisson distribution, though rather differently from me. I'll run the calculations based on my own assumptions to see what comes out.

hivizman

This calculation is based on the reference provided by Iwannahurl, using the assumptions in my earlier post.

In a lottery with six numbers out of a possible 45, there are 8,145,060 possible combinations, of which only one combination wins the jackpot. If 2,000,000 people play the lottery every draw, and each person buys on average 4.07253 tickets, then a total of 8,145,060 tickets are sold, and on average there will be one jackpot winner per draw (in practice, because of duplicate and missing numbers, there will be some draws with multiple winners and other draws with no winners).

The average probability that a particular person will win the jackpot in the lottery is a given draw is 1/2,000,000 = 0.0000005 (5*10^-7) - note that the actual probability for any individual depends on how many different tickets he or she holds in that draw, but the average probability is equivalent to 4.07253/8,145,060).

The distribution of the number of jackpots won by a person in n draws is approximated by a Poisson distribution with mean n*0.0000005. Take n = 2080 (20 years of draws at two draws per week). Then the mean of the Poisson distribution is 0.00104.

The probability that a given person wins no jackpot is e^(-0.00104), which is equal to 0.998960515, the probability that a given person wins exactly one jackpot is 0.00104*e^(-0.00104) = 0.001038919, and the probability that a given person wins two (or more) jackpots is 0.00000054 (same as my earlier post).

The expected number of double (strictly, multiple) winners in a population of 2,000,000 is therefore 1.08. The distribution of double winners is also Poisson, with mean 1.08. The probability of no double winners is e^(-1.08) = 0.3393, so the probability of at least one double winner in 2080 draws is 0.6607.

To have a 0.5 probability of at least one double winner, you require, on the assumptions above, 1666 draws (slightly more than 16 years of twice-weekly draws).

Iwannahurl

I actually didn't know the Lotto had gone 6/45!

hivizman

Iwannahurl wrote: »

I actually didn't know the Lotto had gone 6/45!

Yes, this change was introduced as long ago as November 2006 - see http://www.lotto.ie/Global/Booklets/2/Lottery_Annual_report_06.pdf (page 16).