UK Lottery Maths.

[peter]

There are 50 numbers (ie 1 to 50) to select from in the weekly lottery draw in the UK. From this range the player selects 6 numbers as his/her entry for the draw, for which she pays £1.

How many possible combinations of 6 numbers can be got out of the integers 1 to 50, and if 10 million lines a week are purchased what is the statistical likelyhood of a winning line being drawn in any given week of the 52 week year (ignore the fact that a small number of the lines purchased will replicate the numbers of previous lines purchased for the weeks draw)?

Can we be confident that say, over the course of a year the results of observation of the actual draws will approximate to the mathematically calculated expected results.

The reason I ask these questions is that selling as I do, hundreds of lottery tickets a week, I am always suprised at the fact that people seem to win the jackpot at least most weeks (the likely hood of this happening would have to be closer, the closer the number of possible combinations is to the actual number of lines bought but if there is a vast disparity in these numbers, that the lottery jackpot is won each week becomes more suprising). Another thing I have noticed is that the number of 'rollover' weeks (ie those weeks where the jackpot has not been won the week before and tus the previous weeks money 'rolls over' into the next weeks prize) does not seem to have become more frequent as participation in the National Lottery has dropped (The number of lines purchased now is possibly one quater of the figure in it's heyday a few years after commencement) and this again suprises me. There's probably nothing in it but........

[I'm Murrin]

I thought it was 1-49?

Chance of getting jackpot (6 numbers) from one ticket:
6/49*5/48*4/47*3/46*2/45*1/44 = 1/13,983,816

13,983,816, of course, will also be the number of combinations of six numbers.

If ten million unique tickets are bought, the chance of one of them winning is therefore 10,000,000/13,983,816 = ~71.5%

However, in practice there will be some duplicate tickets since you're talking about 10 million random sets taken from only 14 million possible sets.

(If you were selecting from numbers 1-50, the magic number is 15,890,700.)

[Vraith]

heh...you Brits have GREAT ODDS.
NY State Lotto is 1 in nearly 50 million...but you get 2 numbers per buck, so winning is 1/25million-ish.

The really big jackpot lotteries are more like 1/150million or somesuch.

[I'm Murrin]

IIRC the Irish and Spanish lotteries have the best odds.

Regarding the disparity: the National Lottery is not a for-profit organisation, being state run. Top three places the money goes is: Prizes, charitable causes, government. There's less drive to increase margins than in a private lottery.

[Hashi Lebwohl]

I thought it was 1-49?

Chance of getting jackpot (6 numbers) from one ticket:
6/49*5/48*4/47*3/46*2/45*1/44 = 1/13,983,816

13,983,816, of course, will also be the number of combinations of six numbers.

If ten million unique tickets are bought, the chance of one of them winning is therefore 10,000,000/13,983,816 = ~71.5%

However, in practice there will be some duplicate tickets since you're talking about 10 million random sets taken from only 14 million possible sets.

I confirm the accuracy of these numbers.

If you look at prior lottery results I suspect you will find that some numbers wind up being chosen more often than others. Yes, in the really long run (say, 1,000 years or so) we would expect all numbers to wind up being a winning number in the same amount but this is not the case. Either there haven't been enough runs of the lottery to produce expected results or there is some minor flaw in the method used to draw the numbers. Based on this, if you wanted to increase your likelihood of winning you look at which numbers have come up the most and choose those. No, that isn't a guarantee because past performance is not an indicator of future performance but it won't hurt your overall chances, either.
Many people play personal lucky numbers, birthdays, anniversaries, etc. I recommend not doing this because ultimately this leads to frustration and unconscious distrust of those numbers and the events or people associated with them.

Naturally, the buying of lottery tickets seems to rise as the potential jackpot rises. I know the last few times the large prizes broke through the $150M threshold that people went crazy buying tickets.

I don't mind playing the lottery from time to time on a lark. My gambling preference, though, is always the casino. As strange as it may sound the odds of winning in a casino are significantly better than playing the lottery.

[sgt.null]

math is hard.

[ussusimiel]

IIRC the Irish and Spanish lotteries have the best odds.

The Irish lottery had to increase the number of combination from 6/36 to 6/39 after a syndicate successfully managed to buy all the combinations for one of the draws in 1992. (It's up to 6/45 now.)

I never play it myself, always just seemed like another form of tax to me.

u.

[Akasri]

You are all wrong... according to the email I got the other day, the German lottery is the best and if I'll just send this guy my bank account number he'll guarantee me the winning ticket!

So there! :p

[Vraith]

You are all wrong... according to the email I got the other day, the German lottery is the best and if I'll just send this guy my bank account number he'll guarantee me the winning ticket!

So there! :p

WOW...can you get me that guys contact info?
It'll be great combined with this schloss in Bavaria I just found out I can get dirt cheap.

[sgt.null]

A guy sent me an email telling me I have money in south Africa. I can use that cash to play the lotteries all over the world. I will be so rich, you will all have to love me then.

[peter]

We all loves yaz anyway Sarge!

I've heard before that by keeping a really close eye on all the worlds lottery draws it is possible to choose the ones where if you buy all the number combinations you can guarentee a return bigger than your outlay. Alex Belos in his book 'Alex's Adventures In Numberland' meets a mathmetician who does just that and has his own island in the west-indies. I'll have some of that please.

The figures then would confirm that we could expect roll-over weeks in the ratio of 1:3 which I guess is about what we see.

[Vraith]

I've heard before that by keeping a really close eye on all the worlds lottery draws it is possible to choose the ones where if you buy all the number combinations you can guarentee a return bigger than your outlay

I'm doubt that is true, though I'm sure you can make the odds very good for that, the amount of cash you are risking makes it an unsafe bet unless you are already mega rich.
Because...[unless some lotteries run differently somewhere].
Let's say its a really small lottery. It only costs you 1 million to buy every number. Realize: your ticket purchases, at 1 buck a ticket, have only created a maximum jackpot of about 400K. [or less].
To reach "break even" 2 things have to happen: 1,250,000 other tickets would have to be sold. NONE of those other tickets ALSO has the winning numbers.
It's even worse if you have taxes on winnings...for instance, U.S., the Fed's a chunk...we'll call it 30% for easy math. So your "break even" jackpot isn't actually 1 million, it is 1.3 million. Which means even more tickets have to be sold, which raises the chances of multiple winners.

[[NOTE: I realize you might be continuing the joking, but there is no smiley face, and I actually know someone who thinks it.]]

[Hashi Lebwohl]

The strategy of buying every combination of tickets then waiting for your win not only makes sense but would be quite profitable, presuming you can do it in the first place.

Let's take a look at the current Texas Lotto, picking 6 numbers from 1 to 54. Odds of winning the jackpot: 1 in 25,827,165. The current jackpot prize is $6M annuitized or $4.27M cash payout; since your investment would be $25.8M this isn't worth it.
However, consider the Texas Mega Millions. Draw 5 numbers from 1 to 56 then a "megaplier" number from 1 to 46 to get overall odds of 1 in 165,949,784. The problem here, though, is that the current annuitized jackpot is only $126M so it isn't worth your time. In order to turn a profit you need ((jackpot * (1 - tax rate))/165,949,784) > 1.
Even if you wait until this becomes profitable, the logistics of buying nearly 166 million tickets makes it impossible--at 1 ticket per second this would take 5.258 years...unless you hire 274 people to buy the tickets, in which case they could buy all the necessary tickets if they did nothing but buy tickets for an entire week. On the other hand, 2,735 people could buy the tickets by spending only 2.81 hours/day for 6 days doing nothing but buying tickets and this could be reasonably done. Of course, now you have to split the jackpot 2,736 ways. That sounds like it would water it down a lot but a $200M annuitized jackpot could net each person about $45,000 (after taxes) which isn't bad for about 17 hours of work.

Although plausible in theory, that would still be logisitically impossible so no one will ever do it. No, your odds are still significantly better if you go to the casino and play baccarat, where the house edge is only 1.06% on banker bets and 1.26% on player bets (presuming an 8-deck shoe is used) or blackjack where the house edge under optimal conditions is only 0.44%. (credit goes to the Wizard of Odds for the house edge calculations on those games)

[I'm Murrin]

People actually waste their money on lotteries that have odds of hundreds of millions to one?

[Vraith]

The strategy of buying every combination of tickets then waiting for your win not only makes sense but would be quite profitable, presuming you can do it in the first place.

I'm sorry, Hashi, no matter how you do it you are not guaranteed a good return on investment...you only have a small chance of any profitable return.
If you buy all the numbers with your 166million, you will probably still lose money, because of multiple winners. There have only been a handful of single-winners over 166million in all lotteries combined. [it would actually have to be well over 200mill jackpot to get profit after taxes].
And, if it is the same multi-state mega millions as up here, the multplier does not count for the Jackpot...only lesser prizes.
Heh...it it really worked, Warren Buffet would do it.
This is the exception that proves the rule that you are never wrong.

EDITED TO ADD cuz of above post:
Not only "people" do it...many millions of people do it many times a week.

[Hashi Lebwohl]

I didn't take multiple winners into account, which would definitely water down your winnings. This was a hypothetical situation and a minor thought exercise, not a case study for an actual attempt to game the system.

No, the only successful attempt at gaming a lottery was carried out by a statistician some years ago. She bought a lot of scratch-off tickets then analyzed the winners and losers against the serial numbers printed on the tickets. Once her investigation was sufficiently complete she could look at lottery tickets then tell which ones were winners without having to scratch them.

Warren Buffet's money-making strategy is the time-tested "buy low; sell high", which actually works.

[peter]

(for the reason it sounded so implausible to me I cited my scource. )

[peter]

People actually waste their money on lotteries that have odds of hundreds of millions to one?

(Sorry for the double post Guys, but this came to me during the night long after my last post!)

When you buy a lottery ticket with no real chance of winning, I think what you are paying for, more than the actual winning, is the 'feeling' that you might win. This stays with you alongside the ticket in your wallet and when stuff gets you down, or you struggle to pay a bill etc, the feeling is there just giving you that little bit of hope when you need it. For this service I think the dollar or pound is well spent - as long as you realise that this is what you are buying. The winning, if it happens, is unrelated - it's an additional gift if you like. The problem is that most people don't get this, and many start doing more and more, moving into scratchcards, etc, etc. What they forget is that old thing they were taught at school that nothing times something is still nothing! If you have no chance of winning in the first instance - then five lines of no chance is still no chance, but you have moved into the place of paying more for your feeling of hope than it is worth. (nb to maximise the value of your ticket buy it on the day after the lottery is drawn so as to get 6 days of hope insted of 1. I do not personally do the lottery - these are just my observations and I am aware the logic is flawed and full of holes)

[TheFallen]

nb to maximise the value of your ticket buy it on the day after the lottery is drawn so as to get 6 days of hope insted of 1.

Leaving aside trying to assign values to such immeasurable woolinesses as hope, your advice is entirely wrong, Peter. Statistically, it's easily demonstrable that by far the best thing to do is to buy your lottery ticket as close to the draw time as possible.

Why? Simple - although your odds of winning don't change in the slightest bit, your odds of dying in the intentionally shortened period between you spending your stake and the lottery draw occurring are massively reduced. Solely from your own selfish point of view, one factor meaning that your stake money will have been wasted - i.e. your shuffling off this mortal coil prior to the draw - is then near on obviated.

Of course this doesn't factor in any potential benefits to your inheritors but hey, it's a dog-eat-dog world out there.

[peter]

nb to maximise the value of your ticket buy it on the day after the lottery is drawn so as to get 6 days of hope insted of 1.

Leaving aside trying to assign values to such immeasurable woolinesses as hope, your advice is entirely wrong, Peter. Statistically, it's easily demonstrable that by far the best thing to do is to buy your lottery ticket as close to the draw time as possible.

Why? Simple - although your odds of winning don't change in the slightest bit, your odds of dying in the intentionally shortened period between you spending your stake and the lottery draw occurring are massively reduced. Solely from your own selfish point of view, one factor meaning that your stake money will have been wasted - i.e. your shuffling off this mortal coil prior to the draw - is then near on obviated.

Of course this doesn't factor in any potential benefits to your inheritors but hey, it's a dog-eat-dog world out there.

The logic of your answer is irrefutable - I refute it thus !

( )