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November 15th, 2006

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05:12 pm - It's The Big One!
If you can recognise which of the many people who have doubtless said "It's The Big One!" over the years I like to quote, award yourself considerable smugness.

The EuroMillions lottery draw rolled over for an eleventh consecutive week last week, so a special stipulation comes into play. It cannot roll over for a twelfth consecutive week, so all the rollover funds are guaranteed to be paid out this week. If there's no jackpot winner this time, the rollover fund will be split among the winners of the second prize. This means that there may very well be a situation where more is paid out in prize money this week than is taken in from entry funds. If that wouldn't be a good reason to play the lottery, I don't know what would be.

Of course, the chance that you'll lose is as just high as ever, and if you win one of the smaller prizes, your smaller prize is going to be about the same as usual; it's just that in the unlikely but nevertheless possible incident that you hit the top prize awarded, which does not necessarily have to be the jackpot, then it'll be much higher than usual - and that makes the lottery more worthwhile playing.

In working out whether this is the case - whether your lottery ticket can be expected to win more than it costs - there are a couple of approaches.

One would be to consider the jackpot only. The theory is that you have a 1 in 76,275,360 chance of hitting the jackpot and your ticket costs £1.50, so you should pay when the jackpot exceeds (£1.50 * 76,275,360), or £114,413,040. As £103,732,783 rolled over from the previous jackpot fund and we can reasonably expect the jackpot to go up by no less than the £15,500,000 it increased last week - indeed, the official estimate of £120,000,000 looks entirely reasonable - then this should be the case. However, this approach is inaccurate because should you hit the jackpot then you may have to share it with others; your chance of winning the entire jackpot depends not only on you hitting it but also on everybody else not hitting it. I shall return to this later.

The other approach would be to consider the total sum of entry fees taken in and the total sum of prize money paid out. We can, for the purposes of this exercise, reasonably ignore the fact that the tickets cost £1.50, €2 or CHF 3.20 depending upon where they are bought, because an appropriate adjustment is made to the size of the other prizes accordingly. Therefore, let us assume that t tickets are sold this week. The total entry fee taken in is E, where E = £1.50 * t .

By paragraphs (F)(2) and (F)(5) of the rules, 42% of the entry fees taken, plus any rollover funds, will be paid out as prize money in that draw (with another 8% of the entry fees taken to form a reserve fund to ensure that the jackpot next time starts at a respectable amount). Accordingly, the total prize fund paid this time is P, where P = (0.42 * £1.50 * t) + £103,732,783 .

For the total prize paid to exceed the total entry fees taken we need P > E or
(0.42 * £1.50 * t) + £103,732,783 > £1.50 * t which, subtracting (0.42 * £1.50 * t) from each side, means that
£103,732,783 > £1.50 * t - (0.42 * £1.50 * t) which, rearranging, means that
£103,732,783 > 0.58 * £1.50 * t and so, dividing both sides by 0.58 * £1.50, means that
£103,732,783 / (0.58 * £1.50) > t or
£103,732,783 / (£0.87) > t , which requires that, as t must be a whole number,
t is no higher than 119,233,083.

So there we have it. Playing the EuroMillions this week will have +EV (i.e., will be worth the money in terms of prize money alone) if no more than 119,233,083 tickets are sold.

We can estimate the number of tickets sold last week, though this estimate will be slightly inaccurate because we can no longer strictly ignore the currency exchange rates. From aforementioned paragraphs (F)(2) and (F)(5), 11% of the cost of each entry goes towards the jackpot fund for that week; we know that last week, the jackpot fund rose from £88,344,099 to £103,732,783, a rise of £15,388,684. As this represents (11% * £1.50) times the number of entries, we can estimate the number of entries last week as having been around 93,000,000. Looks promising!

However, we have some better data than that. Last time the most comparable situation occurred, an eleven-time rollover on 3rd February 2006, £105,412,292 of jackpot had rolled over to that draw, three jackpot prizes of the then-equivalent of £42,019,877 were paid out, implying somewhere around 125,000,000 tickets were sold for an eleven-time rollover draw such as next week's.

There are competing reasons why the number of tickets sold this time may be different from 125,000,000. It should be noted that last time, the previous week, around 106,000,000 tickets had been sold, compared to 93,000,000 this time, which would imply that ticket sales are down by about 12%, implying that the sale this time might likewise be about 12% down on 125,000,000 - a guess of 110,000,000 or so might seem reasonable.

However, that eleventh-time rollover did not have the stipulation that the jackpot would roll over to lower prizes if necessary; the weird paragraph (F)(6)(b) of the rules suggests that the jackpot would have rolled over a twelfth time and been guaranteed to be paid out on a thirteenth draw, which is no longer the case. Estimating the effect of the stipulation coming into play for the first time is a matter of sheer conjecture, but if people like Richard Lloyd are prepared to play EuroMillions for the first time this week then it does seem reasonable to suggest other first-time players will come out of the woodwork. If we pluck an increase of 5% (or, at least, between 2% and 8%) out of thin air then it seems reasonable to estimate between 112,000,000 and 120,000,000 tickets can be expected to be sold, which makes the EuroMillions ticket borderline +EV.

All that said, we can exploit the information that is available even better still! It is established protocol that Camelot, the National Lottery operators in the UK, have been known to revise their estimate of the jackpot closer to the draw, depending upon how sales have been going; they have privileged information about the number of tickets that have been sold which they will use to recalculate their estimate. If 119,233,083 tickets are sold then the jackpot fund for this draw can be expected to be £123,406,241.

Accordingly, my overall conclusion is that if the estimated jackpot rises to £124,000,000 or higher then it seems reasonable to assume that people who know expect more than 119,233,083 tickets to be sold and so the EuroMillions ticket is no longer +EV; if the estimated jackpot rises to £123,000,000 or less than it seems reasonable to assume that people who know expect more than 119,233,083 tickets to be sold and so the EuroMillions ticket remains +EV. The current estimated jackpot is £120,000,000, and there seems to be no incentive for the estimate to be artificially high or artificially low, so I would have thought that a EuroMillions ticket is currently +EV.

Other questions that present themselves:

1) How large is my chance of winning at least a hundred million quid? As discussed, your chance of winning the entire jackpot depends not only on you hitting it but also on everybody else not hitting it. The latter factor depends on the number of entries, but suffice to say, if there are (e.g.) 119,233,083 entries, including one of yours, then the chance of the other 119,233,082 entries all missing the jackpot is (76,275,359/76,275,360)119,233,082, which is (assuming sufficient accuracy in my calculation!) about 20.9%. Thus the chance of your single ticket hitting the jackpot remains 76,275,360, but of scooping the entire jackpot is about 1 in 364,000,000. It's impossible to be more accurate before knowing exactly how many entries there will be.

2) What happens if nobody hits the jackpot this time? ...which, as discussed, is something like 20% likely. The £103,732,783 rolled over goes along with this draw's addition to the jackpot fund and another 3.7% of the entry fees of this draw to form a pool of something like £130,000,000 to be split among all the people with all five main numbers right and one lucky star. As the chance of getting that result is about 1 in 5,448,240, one might expect that around 22 tickets might come into that category and would each win something like £6 million if nobody wins the jackpot or something like £300,000 (could be a quarter of that, could be four times as much) if anybody hits the jackpot.

3) Should I buy more than one ticket? If you believe this is a +EV bet, then, massively simplifying, the Kelly Strategy suggests you should bet Bankroll * EV / Variance on EuroMillions. I'll let somebody else calculate the variance of a EuroMillions ticket, but the EV / Variance ratio is going to be pretty damn small, so - for most intents and purposes - "no". In truth, you shouldn't be buying a whole EuroMillions ticket, you should be buying a tiny fraction of one, but it makes sense to round this up to one to exploit the +EV at all.

4) But what about the fact that some money goes to charity? On the assumption that 28% of your entry fee goes to good causes, if you're prepared to assume you're ambivalent between giving (28% of £1.50) to charity and buying a EuroMillions ticket, then using the language above, we need a P/E ratio of over (1 - 28%), so
((0.42 * £1.50 * t) + £103,732,783) / (£1.50 * t) > 0.72 , so, rearranging,
0.42 + (£103,732,783 / £1.50 * t) > 0.72 and hence
(£103,732,783 / £1.50 * t) > 0.3 which can be rearanged to give
(£103,732,783 / (£1.50 * 0.3)) > t with the happy conclusion that a ticket will be +EV if there are fewer than 230 million tickets sold, which will be the case unless the estimated jackpot exceeds £141 million. Seems likely to me!

ETA: Didn't win the jackpot. In fact, nobody did. As hinted at in 2) above, 20 people each won the 5+1 prize, which worked out at around £6½ million or €9.65 million each, which is a pretty handsome chunk o' change. Happily, 7 of these 20 winning tickets were British, though only something like 15%-25% of the entries normally are. (Apparently this time we bought second most tickets behind only France, and France only had 4 big winners to our 7. Hurrah!) I think my assumptions about exchange rates may have been a simplification too far. The number of tickets bought was probably between 120,506,199 and 134,293,684 and the jackpot pool ended up being £123,232,395, which (if anything) looks a little low. Hopefully more detailed figures released in coming days - possibly requiring a country-by-country breakdown, alas - will permit more detailed analysis.
Current Mood: geekygeeky

(25 comments | Leave a comment)


[User Picture]
Date:November 15th, 2006 06:09 pm (UTC)
This just seems silly to me, at least from the POV of, well, everyone.

We have a game here called "Mega Millions" in the state. To win the big horkin' grand prize, you must successfully match 5-of-56 numbers, and 1-of-46. Odds are roughly 1 in 176 million. Knowing this, and being able to compute expected value, I know that rationally, I shouldn't buy a ticket unless the jackpot is higher than $176,000,000. I haven't checked the window of my local convenience store, but it hasn't been $176 million in a long time.

When you consider that the jackpot starts at $12 million, and only starts to pick up steam after several weeks, when the jackpot will increase by several millions of dollars; not like "Eggheads," where it's 1,000GBP every show.

So, I'm not playing their game. If they were to put a ceiling on the prize money that says it stops rolling over at, say, $150 million, I'd never play the game.

I've conveniently forgotten all of that spiffy math that you've done because it's 10 in the morning here and I'm running on six hours sleep. But I do enjoy a good chat about probability. So thanks for that either way.
[User Picture]
Date:November 15th, 2006 09:43 pm (UTC)
Oi, oi! You're taking exactly the first approach by (a) considering the jackpot only and (b) not considering the possibility of other people sharing the jackpot with you.

There was another lottery game in the UK, Lotto Extra, which paid out a rolling jackpot for six correct 1-49 numbers only. Jackpots there started at a million and crawled up at a terribly wimpy pace - a couple of hundred thousand here, a couple of hundred thousand there, the only people playing it presumably being the ones who were obsessive about playing their lucky numbers for fear of the instance that the time they didn't play their numbers, the numbers would come up. Predictably, once the jackpot eventually climbed to £5-6 million, it started to increase more quickly, but it never reached very much. The game was closed due to lack of interest, eventually; I wish I had spotted the last week the game was to be played, with all the rollover funds presumably to be distributed somehow or another, which would very likely have been +EV itself.
[User Picture]
Date:November 15th, 2006 07:25 pm (UTC)
Prayer might help. :-) So, all that and your advice is to definitely buy one this week, but only one. Gotcha!

Although, wouldn't playing the same numbers every week improve your chances over time? No, nothing that has occurred before effects a future gambling event. Or, does it matter?

Besides it's only 1.5 pounds...cough it up. Send me a postcard from da islands mon!
[User Picture]
Date:November 15th, 2006 09:15 pm (UTC)
No, it doesn't really make any difference which numbers you play, so you may as well get a 'lucky dip'.

There is a possible effect that I wonder about, which is that maybe other people choose numbers based on dates that are important to them, which will skew the possible prizes somewhat, but I don't know if it really happens enough to make a difference.
[User Picture]
Date:November 15th, 2006 09:45 pm (UTC)
It's definitely a question worth asking. One approach to picking numbers which are relatively unlikely to be duplicated is discussed there; presumably similar procedures could be drawn up for EuroMillions and the other games as well.
[User Picture]
Date:November 16th, 2006 03:04 am (UTC)
Well, considering the factor that some people 'pick' them numbers (they do here, while others can just let the computer pick), the more 'odd' the combinations the more likely you'd be to win it by yourself, even if the odds didn't change to win it at all. Say you picked 1-2-3-4-5. It is both unlikely that a computer would do it or another player would either. Sure, you might never win, but wouldn't you have the same odds as picking some random, spread-out numbers or family members birthdays? :)
[User Picture]
Date:November 16th, 2006 09:38 am (UTC)
It's a popular factoid (ie quite possibly totally untrue) that quite a lot of people pick 1-2-3-4-5 or similar, precisely because they think nobody else will...
[User Picture]
Date:November 16th, 2006 10:23 am (UTC)
I very vaguely remember (i.e. might be making this up, but I don't think I am) that it was once announced that 30,000 tickets play 1-2-3-4-5-6 on the National Lottery each week - almost certainly fewer than that, these days.
Date:November 16th, 2006 05:51 pm (UTC)
I think I told you that once, and from memory the correct value is around 10,000 to 12,000 people according to Camelot (see also http://lottery.merseyworld.com/Info/FAQ_tickets.html)
[User Picture]
Date:November 16th, 2006 04:54 pm (UTC)
Well! There goes my castle in Normandy!!!! :-) Will have to settle for a share of the prize and a one room flat in E. London!!! ;)
Date:November 15th, 2006 10:48 pm (UTC)
Again, it isn't really better or worse than any other method, but as a Lucky Dip entry is essentially asking two independent random number generators to produce the same set of numbers... you'd really need it to be an extraordinarily lucky dip!
[User Picture]
Date:November 16th, 2006 09:40 am (UTC)
That has to be wrong, but it's not 10 yet so I can't explain why. Maybe this afternoon.

I have an Excel spreadsheet that does [bogus] stats on the National Lottery, but I do Lucky Dips on the Thunderball and Euromillions.
The Lucky Dips are ahead, so far.
[User Picture]
Date:November 16th, 2006 09:35 am (UTC)
I think that it's been demonstrated that date-like numbers (1-12 and 1-31) do get chosen significantly more often, and that avoiding them will make a significant difference to your payout. I don't have a reference for that though.
[User Picture]
Date:November 15th, 2006 09:46 pm (UTC)
Should I hit the jackpot (and, to be fair, I haven't bought a ticket yet - I'm waiting to see how the jackpot estimates move) then it seems likely that the whole of the Friends list might be getting something more than a postcard. Likely, not definite, mind you ;-)
[User Picture]
Date:November 16th, 2006 03:06 am (UTC)
Just how cheap is that? If you win 105 pounds, you can't even promise some Cadbury? Nada!?!?!? :P
Date:November 15th, 2006 10:54 pm (UTC)
Given 3) above, can I buy a 1/150 share of your ticket if you get one? :-)
[User Picture]
Date:November 16th, 2006 12:05 am (UTC)
Yes, but only on the basis that I am not PayPal-ing you 4p if I win bottom prize!
[User Picture]
Date:November 16th, 2006 03:07 am (UTC)
105 million pounds or bust eh? LOL
Date:November 16th, 2006 03:46 am (UTC)
Hehe all that calculating confuses me :)

But I made myself a tool to monitor my Euromillions combinations and mail when I win, cause I always forget to check it or renew my ticket in time.

http://www.lottomate.org if anyone is interested
[User Picture]
Date:November 16th, 2006 09:42 am (UTC)
If you enter online via http://www.national-lottery.co.uk/ they'll email you if you win, so you never need to worry about it.
[User Picture]
Date:November 16th, 2006 07:40 am (UTC)
Funny, here in California "It's The Big One" has an entirely different meaning.

Although the last "Big One" was in 1906...
Date:November 16th, 2006 05:54 pm (UTC)
What would Phil Helmuth do?
[User Picture]
Date:November 17th, 2006 08:54 pm (UTC)
(coughs, takes deep breath)

Date:November 17th, 2006 03:28 pm (UTC)
I don't understand point 3). If a bet is +EV, then ten such bets should be multiple EV, if not quite 10 x EV.

In other news, I appear to have already won the EuroMillions:


Euro Millions Lottery,
Gregorio Herrraez 15

(i)the file Ref number: Ef/04679/SA
winning numbers:8-16-19-43-45-1-4
(iii)email ticket number:754/22/76
(iv)Batch Number 620-112-934-11

In view of the yearly sweepstake of
the above named organization held
on the 3rd of August 2006. It is
my pleasure to inform you that your e-mail address attached to the above ballot No came up in the third stake.
This invariably means that
you have emerged as the prize recipient in the third category with an allocated sum of 870.812,79 euro (Eight hundred and seventy thousand, eight hundred and twelve euro,seventy-nine cent)only in the open ballot device.
Be informed that all participants were selected from a random computing ballot system...
For immediate release of your cash prize, kindly contact our Paying Agent.

Mr.Sabastin Antonio
C/Santander, 1 28008

The above claim agent will assist you in the processing and remittance of your prize funds to you,note that not later than one week After this date if you do not contact and process your winning prize, all funds will be returned as unclaimed

Congratulations once again.

Yours Faithfully,

Rita Maria
(Euro Millions Lottery)
[User Picture]
Date:November 17th, 2006 08:49 pm (UTC)
The best analogy I can think of is this: suppose you have won Play Your Cards Right/Card Shark(s) and have got to the cash cards at the end. Your first card (*) is a 6. Obviously you would choose to go higher than a 6, but how much do you bet? Why do you bet the amount you do? Is there a correct amount to bet? Should you bet your entire bankroll every time on 6s, 7s, 9s and 10s?

(*) ...ignoring the possibility of being able to change the card - perhaps you have changed a 6, only for it to be replaced by another 6. ("Oh dear, the cards are being funny tonight!")

If you've already won it then damn, I needn't have bothered playing :-/

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