Taking into account the probabilities for each prize class, it is straightforward to compute theoretical payouts (assuming all 139,838,160 combinations were each played exactly once, i.e., a total stake of €167,805,792 at €1.20 per tip: 139,838,160 × 1.20 = €167,805,792.00). In general, only half of total stakes, i.e., 50%, is distributed as prizes, so €83,902,896.00. For further calculation you need, in addition to the probabilities above, the payout shares per prize class. Because the lowest prize class has a fixed payout (€6.00), the calculation for LOTTO 6aus49 proceeds in two steps:
1) Compute payouts for prize class 9 (lowest class, two correct plus Superzahl) and 1 (six correct plus SZ, the top class, i.e., the jackpot). First, total prize money for class 9 is number of winners × €6.00. In addition, 15% of the overall prize fund goes to class 1.
2) Compute payouts for classes 2 to 8. The remaining distributable pool is allocated as follows:
Class 2 (6 correct) = 15%, Class 3 (5 correct + Superzahl) = 5.2%, Class 4 (5 correct) = 15.5%, Class 5 (4 correct + SZ) = 4.3%, Class 6 (4 correct) = 10.2%, Class 7 (3 correct + SZ) = 8.7%, Class 8 (3 correct) = 41.1%.
In each draw the payout per prize class depends on the number of winners in that class. Hence, since only 50% of stakes are paid out, the following theoretical payouts result (under the assumption that all combinations were played once, there are exactly 1,851,150 winners in class 9; that class therefore requires 1,851,150 × €6.00 = €11,106,900. For classes 2–8 there remain €83,902,896.00 − €12,585,434.40 − €11,106,900.00 = €60,210,561.60 to distribute):
Prize class 9, 2 correct numbers (of 49) with the correct Superzahl:
Fixed payout of €6.00 (this class needs 1,851,150 × €6.00 = €11,106,900)
Prize class 8, 3 correct numbers (of 49) without Superzahl:
=60,210,561.60×41.1%×(1/2,221,380) = 60,210,561.6×0.411/2,221,380 ≈ 11.14 ≈ €11.10
Prize class 7, 3 correct numbers (of 49) with the correct Superzahl:
=60,210,561.60×8.7%×(1/246,820) = 60,210,561.6×0.087/246,820 ≈ 21.22 ≈ €21.20
Prize class 6, 4 correct numbers (of 49) without Superzahl:
=60,210,561.60×10.2%×(1/121,905) = 60,210,561.6×0.102/121,905 ≈ 50.38 ≈ €50.30
Prize class 5, 4 correct numbers (of 49) with the correct Superzahl:
=60,210,561.60×4.3%×(1/13,545) = 60,210,561.6×0.043/13,545 ≈ 191.14 ≈ €191.10
Prize class 4, 5 correct numbers (of 49) without Superzahl:
=60,210,561.60×15.5%×(1/2,322) = 60,210,561.6×0.155/2,322 ≈ 4,109.22 ≈ €4,019.20
Prize class 3, 5 correct numbers (of 49) with the correct Superzahl:
=60,210,561.60×5.2%×(1/258) = 60,210,561.6×0.052/258 ≈ 12,135.46 ≈ €12,135.40
Prize class 2, 6 correct numbers (of 49) without Superzahl:
=60,210,561.60×15%×(1/9) = 60,210,561.6×0.15/9 ≈ 1,003,509.36 ≈ €1,003,509.30
Top prize class 1, 6 correct numbers with the correct Superzahl:
=€83,902,896.00×15%×(1/1) = 83,902,896.0×0.15 = ≈ €12,585,434.40
