Teesside Snog Monster (jiggery_pokery) wrote,
Teesside Snog Monster

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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.
Tags: gambling, games, lottery, mathematics, probability
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