joy wrote:And if lottery ticket actually worked, investors would buy them.
Actually, there is a rare situation where they do work, but an investor needs to buy all of the numbers, so this is not an investment for mom-and-pop. Of course, if you had €10M , you could be a corporate raider instead, in which case, opportunities are more frequent.
The rare situation is where the "true" prize (usually only half of the stated prize, but that state-sanctioned lying is a separate topic) is greater than the cost of buying every ticket and is greater by enough to deal with the possibility of having to share. Seat of the pants calculation: Suppose there are 45 possible numbers and you have to pick 6 of 6, but they can be in any order. That's 8,145,060 different tickets. Suppose each ticket is €1 (and suppose that you can somehow order tickets electronically so you can get them all) and the jackpot is €20M. You buy a guaranteed win for at little more than €8M. Now, big question is whether you'll have to share. That all depends on two things (all others things can be ignored, because this is "seat of the pants") and those are: 1) how many other buyers there are, and 2) whether the outcome is 1-2-3-4-5-6. In the latter case, you'll lose most of your money because you'll be splitting it 100 ways because so many people pick inane sequences - fortunately, there is less than an 8M-to-one chance of that. How to tell how many other buyers there are? I don't know, but you might infer it from the difference between the last jackpot and the current jackpot. For example, if last month's jackpot was €15M and rolled over, and this month's is €25M, figure half the money gets siphoned off by the operators, so that means 20M tickets were bought for this month. (This totally ignores the smaller prizes, but since I am ignoring them in the expectation value of the win, that cancels out. I think.) Of those 20M tickets, 8M were you and 12M were other-than-you. Assuming a random scattering of their numbers (not a valid assumption) over the 8M possible numbers, there is a 23% chance that every one of those other 12M tickets are not the winning numbers. In such a case, you would have a 23% expectation of €25M, some similar expectation of €25M/2, etc.
Of course, the more the jackpot rolls over, I guess the more people would buy, so it would be rare that the particular numbers would occur. I didn't think the stars would ever align this way, but apparently they did, at least once, IIRC, and the investors made a little bit of money, but not a lot.