Probability of EuroMillions

The Euromillions lottery is a two panel game so that a player chooses 5 numbers from a first panel of 50 numbers and 2 numbers from a panel of 9 numbers. For a single ticket entry, the probability of winning each of the twelve prizes is as follows.

Tier 1, 5 main balls and 2 lucky stars

The first term represents the calculation for the probability of getting 5 main balls correct: is the number of ways of selecting 5 of the 5 winning numbers, is the number of ways of selecting 0 of the 45 non-winning balls and is the number of ways 5 numbers can be chosen from 50.

The second term represents the calculation for the probability of getting 2 lucky star numbers correct. These terms can be multiplied together because they are independent events – the drawing of the main balls has no influence on the drawing of the lucky star numbers.

As for the UK Lotto system, in terms of probability of winning prizes and frequency of winning prizes, the e-lottery system for Euromillions lottery reduces down to the same as purchasing 36 random distinct tickets. There will be draws when the system numbers are not drawn as winning numbers.

On these occasions 36 tickets with more varied numbers may win more prizes (and money) than the system tickets. However, on weeks when the system numbers are drawn, there will be many prizes won. Over the longer term, the two alternatives will be equivalent in that the average number of prizes won will be the same.

The difference observed, that on some weeks the system wins very few prizes and on others it wins many, is a consequence of the difference in variance in return of the alternative systems. The system puts “all its eggs in one basket” – it either wins many prizes or very few. The alternative, 36 random distinct tickets will have the same average number of wins over the long term, but may perform less erratically.