This topic is close to my heart, since I majored in Statistics in the university. Your explanation is quite clear and concise. Let me add my 2 cents worth. Hopefully it won't scare away any mathaphobes (is that a word?).
When you multiplied the probabilities, you're assuming that the two events are independent. In this case, the choice of 5 out of 50 numbers and 2 out of 10 numbers is assumed to be independent. If that were not the case, your probability would be different. I can't calculate this new probability because I'm not familiar with the Eurojackpot. Also, you can only add two "OR" events when they are mutually exclusive, i.e. they can't happen at the same time.
Philosophically, most people feel an emotional high when they buy a lottery ticket, hoping they will win. Then they get disappointed when they lose. Many people like this emotional roller coaster, which is why they buy lottery tickets like candy. Contrasts in emotions cause people to feel them more acutely. This is true in many aspects of life.
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Those events are independent and therefore must be calculated that way :) Thank you for your input about this and for sharing your thoughts with us. The emotional high of winning or losing is kind of the point here but I hope that not many people feel that disappointment when they do not win. It is just a game, can bring a lot but still just a game and should be treated as such.