### Onsdagslotto – when to play and win the superpulje

The so-called “superpulje” in “Onsdagslotto” (Viking Lotto) has reached a record breaking amount on 125 millions and with extra so-called jackpots reaches 157 millions. This is around 30 million US Dollars or 22 million Greek Euros (equivalent with 22 million German Euros).

It is dangerous to talk about the statistics of Onsdagslotto as twice our local statistics watchdog Mikkel N. Schmidt has caught researchers giving the wrong odds: The first time blogging Mikkel caught Jørgen Hoffmann-Jørgensen from University of Aarhus giving the wrong odds. The second time Master Schmidt found that University of Copenhagen Professor Mogens Steffensen‘s odds or the Politiken newspaper reporting his odds were wrong.

Fearless of Mikkel I will now attempt my computations (which are probably wrong).

In Onsdagslotto you pick six numbers from 48. The number of different combinations/rows are (48*47*46*45*44*43)/(6*5*4*3*2*1) = nchoosek(48,6) = 12’271’512 = around 12 millions. The so-called superpulje is released if an extra independently picked number (a 7th number) hits one of the six numbers. The probability that the superpulje is released is thus 6/48 = 1/8. It means that on average you need to play nchoosek(48,6)*8 = 98’172’096 = around 100 millions rows before you win the superpulje.

Hoffman gave (48*47*46*45*44*43*42)/(7*6*5*4*3*2*1) = 73’629’072 for the superpulje. This number is correct if you were to hit 7 numbers from the 48. But this is not how the rules are (as far as I understand).

In the report from Professor Mogens Steffensen the newspaper made it sounds as if the value of 98’172’096 was the number of combinations from one coupon. But there are 10 games on each coupon, so the average number of coupons you need to play are nchoosek(48,6)*8/10 = 9’817’209 = around 10 millions, – as Mikkel notes.

Apparently, it costs 4 Danish Kroner (DKK) to play one row/combination/game. To play all combinations will cost you 12271512*4DKK = 49’086’048 DKK = around 50 million DKK. Peter Brodersen noted that as the superpulje was hit this week you could have gain a considerable number of money if you had played all combinations. The bad news for the strategy of playing all rows is, however, as Brodersen also mentions, that you do not know if the seventh number hits the 6 others and you do not know whether you need to share the amount with other players.

On average you need to spend nchoosek(48,6)*8*4DKK = around 400 million

DKK playing all combinations before you hit the superpulje. It seems to be more difficult to compute the probability that you have to share the amount from the superpulje. During these times with a large superpulje Danes are playing for around 65 to 82 millions DDK each round, meaning around 20 million combinations are played in Denmark and that on average each combination is played around one or two times (82000000/4/nchoosek(48,6)). However, the superpulje is shared with other countries in the Nordic region. One blogger notes that we are 33.4 millions in the region. If the people in the other countries play at the same rate as Danes, does that means that a superpulje winner has to share it with 9 other people on average? (33.4/5.5*82000000/4/nchoosek(48,6)=10.145). I am not sure I understand the rules correctly… Because last week when the superpulje could have been released no player hit the six correct numbers among the 48. If Danes are alone to play the probability of no-one not hitting the six correct is around 25% (1-1/nchoosek(48,6))^(65000000/4). Whereas if we use the Danish playing rate on the entire Nordic region population we get around 0.0003 (1-1/nchoosek(48,6))^(33.4/5.5*65000000/4). So either I am computing this wrong or I misunderstand the rules or the playing rate is quite lower in the other countries, – or it was a very unusual drawing. Yet another explanation is that some people play systematically. It has been reported that one particular combination was played 1’600 times. If you win the superpulje alone you will apparently also receive the secondary prizes, – if I understand correctly. That amount I read on one news site (Avisen.dk) to be around 10 million DKK.

The rate of which other people play, their rate of systematic playing and the secondary prizes make the computation of when it is an advantage to play difficult. If you disregard the secondary prizes it seems that the superpulje needs to grow to at least 400 million DKK before it is an advantage to play “against” it. It needs to grow further if you count in the other players that might hit your six numbers.

One popup page on the Danish lottery website states that the average payback percentage is 45%. It is unclear for me how the payback is distributed between the different prizes. If we assume that 40% is used for the secondary prizes it means that if we play for 300 million DKK we will on average get 120 million DKK back from the secondary prizes, the rest, 180 million DKK is at stake for winning the carried-over superpuljen (if my understanding is correct). Given that the amount accumulated in the superpulje is now going towards the 180 million it seems that it is almost an advantage for me to play, – provided that the rest of you do not play so I have to share the prize.

*(Correction: Typo 17:39)*