If a ticket costs $1 and you have a 1 in 10 million chance of a win, any payout larger than $10 million (after taxes) gives a positive expected return. Unwise to invest your life savings on a longshot, but still worth buying a few tickets if the expected return exceeds the market rate. I believe the Kelly criterion gets you to an exact percentage of your wealth to spend on tickets.

> any payout larger than $10 million (after taxes) gives a positive expected return.

and conditional on you being the sole winner too (unless the lottery is guaranteed to only have one winner).

Taxes, too: the government will be taking over a third of your winnings for a big payout.

depends on the country

Discounting the possibility of multiple winners splitting the jackpot, and not taking into account taxes on the earnings.

Sure, if you take the whole population, then each ticket will, on average, be worth more than $1. That doesn't change the odds of my making money back on this. There's going to be a handful of people who massively skew the statistics, but it doesn't mean that for the vast majority of players the odds get any better or that their ticket becomes more winning. Expected return doesn't matter on a ticket by ticket basis when the odds and winnings are so skewed, it creates an illusion of worthiness that doesn't exist for the single player.

Games of chance have an expected value [0], in this case the expected amount you can win with your ticket. Normally, this is very low, lower than the price of the ticket, for sure. However, when the payout is high enough, the EV goes up. If the EV surpasses the price of a lottery ticket, that's the point where you think about "investing".

Of course, calculating EV is very hard, so people just draw a line in the sand - "if the payout is $50MM or over, I will buy a ticket, otherwise, I won't".

0 - https://en.wikipedia.org/wiki/Expected_value

I think the problem with that is it forces the data into a normal distribution when it's not actually the case, and instead it's a very large bump at one end with an incredibly long tail that drags the other tickets up?

It makes sense in statistics, but as a single person with a single ticket, it's much less relevant. The EV of a ticket is over 100% of the cost, but that doesn't matter for 99.9999% of the players and only matters for the very very few with the correct numbers.

Another thing about EV that gamblers conveniently forget is that it only matters if you are repeating the experiment many times. So if the lottery odds are 1:100M, and you're only buying one ticket, your odds are effectively zero, and it doesn't matter if the payout is $200M, $500M, $1B or $10B.

The casino can live with a 0.01% edge because they are conducting millions of games. You're not playing that many times.

I think, it's meant like this:

If the lottery gets too big, then it's an investment = if your gambling gets to big, then it's an investment. If you do big gambling, then it's an investment. The law of bigger figures :) the often you try, the higher the chances (whereas the chances stay unchanged here)

It's not about the lottery and it's pots itself (like euro jackpot, USA Powerball.. etc)