Building on my game above, suppose I told you that instead of paying me $2 for a 0.0000001% chance to win, you could pay me as much as you want ($x) and your chance to win the $2.1 billion would be given by (x/2 * 0.0000001)%, **up to a maximum of a 50% chance**. Under this new game, if you pay me $1 billion, you have a 50% chance to win $2.1 billion and a 50% chance to walk away with nothing. The odds of 50% aren't bad, but you're risking $1 billion. This is unlikely to be worth it unless you are some sort of institutional entity with a lot of money to blow.

Except in your game you're not even considering the limits on possible number combinations. You could purchase every single # combo possible (if it were feasible) and you are guaranteed a win. Not a 50% chance.

This was handled in my construction of the problem. Note the bolded part above where I specifically said that no matter how much you pay me, your chance of winning does not exceed 50%.

If your point is just that my example is different from a real lottery, I was aware of that. However, the same issue is present in the real lottery. In the real lottery, if you buy every number, you are guaranteed a win, but you aren't guaranteed a profit because the pot could be split arbitrarily many ways. It then becomes a question of whether it's worth risking the insane amount of money required to buy every ticket even though that procedure is not guaranteed to yield a profit. As you'll recall, that's the same issue you faced in my example.

To restate both points of

my original post, here is why a slight expected value does not mean that it is rational to purchase a lottery ticket:

- If you buy a small number of tickets, your chance of winning is approximately zero regardless of the expected value. Therefore, a positive expected value does not mean it is rational to buy a small number of tickets.
- If you buy a large number of tickets, you can engineer a high chance of profit, but at the risk of losing a massively huge investment. This is unlikely to be a rational trade for non-institutional entities (and usually not even for institutional entities).

My simplified lottery illustrated both points. Those points also apply to the real lottery (except that the real lottery is unlikely to have a positive expected value in the first place). My core thesis remains true: a positive expected value does not mean that it is rational to buy a lottery ticket. The person I responded to above seemingly believed that the expected value of a ticket was the main determining factor in whether it is rational to purchase one. That is very wrong.

...It's probably close to being +EV.

The whole point of my posts in this thread is that

*even if that is true*, it was still totally irrational to purchase a ticket. It was not in any way mathematically justifiable, regardless of the alleged expected value. Relying on expected value as the sole predictor of whether it is rational to purchase a lottery ticket is like relying on AAPL stock as your sole store of value for retirement savings: it should be part of the calculus, but only a small part.

Expected values by themselves are a good guide to rationality only in certain narrow circumstances, usually involving a game that (a) can be played repeatedly with the same odds each time, (b) requires only a small investment for each attempt, and (c) offers pretty decent chances of winning on any one attempt. In other situations, expected value has less relevance to rationality. In the context of the celebrated powerball drawing, expected value had pretty much no relevance to whether it was rational to buy a ticket.