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JLC,

So about 60% of the time it is single winner.

More than I'd expect, but I also think the trend is towards many people only playing on the big jackpots, which should decrease the number of single winners. Could be wrong about that, though.

I'm sure that can easily be factored into the statistical formula.

I'm still not sure the standard formula applies here. Let's consider, like you did earlier, a really simple lottery, but let's simplify even more. Let's say you pay \$1, pick a number from 1 to 10, and the lottery picks a single number from 1 to 10 for the winner. For simplicity, let's say no one else plays. If you buy ten tickets, each with different numbers, you should win once at each lottery drawing. If the prize is, say, \$11 then you have a "positive expected return". With a prize of \$10 or less you don't.

But... now let's divide that into ten new scenarios. In scenario 1, you're only allowed to buy 1 ticket in your entire life. In scenario 2, you're allowed to buy 2 tickets in your life. And so forth, up to scenario 10 where you can buy ten tickets. Given a prize of \$11 again, what is the "expected return" for each of the ten scenarios? In scenario ten, if you buy all ten tickets in the first lottery your return will be exactly \$11 for a bet of \$10. But in scenario one, your return will be either exactly \$0 or exactly \$11 for a bet of \$1... and the \$0 return is far more likely than the \$11 return. And in scenario two, your return for a bet of \$2 could be \$0, \$11, or \$22 (if you play one ticket per lottery) or it could be \$0 or \$11 (if you play both tickets in the same lottery drawing).

I understand the classic definition of "expected return", but I don't think it has any meaning with respect to the previous paragraph, because the limitations are important criteria which are ignored by the classic definition of "expected return".

Similar limitations apply to the actual lottery in real life. You can't possibly buy any significant fraction of the total number of possibilities with any reasonable amount of money even if you play every lottery for the rest of your life. And, with respect to the jackpot, you either win or lose (nothing in between) in each drawing you enter. Therefore, there really is a limitation on both the number of tickets per lottery played and on the number of lotteries played, compared to the number of possible combinations in the lottery. (Note that even if you had the money and wanted to buy every possible combination, you couldn't. It would take far too long just to print the tickets out.) And as I just showed, it matters what the ratio is to the number of possibilities and the number of attempts than can actually be made. The classic "expected return" is irrelevant when the ratio becomes extreme... as it is in the big lotteries. Some different formula is required, if the result is to have any useful meaning.

Phil