I think ZCT’s answer is quite good. I may not be able to improve on it. But I think I can still add meaningfully to the discussion.

I never play lottery games, but I no longer think that the straight forward mathematics of comparing price to expected value makes a strong enough argument against playing.

Let’s not forget, many wise economists have come to recognize that the value of money is relative, and highly subjective. That is to say, $10 is worth more than $1 in every case, but it is not necessarily worth 10 times as much. In some cases it can be worth significantly more than that to a particular individual. In most cases, it’s worth less than that. This might be a jarring fact to confront for the most mathematically inclined, but consider that money is only worth what you can buy with it.

On the one hand, a person who has nine apples usually does not value an additional apple as much as a person who has no apples.

On the other hand, certain critical values can only be obtained if you meet a bottom threshold on price. It certainly will take many thousands of dollars at least, to cross the threshold from homelessness to secure private housing. It might take a half-million to go from working stiff to capitalist, or it might take many millions to get a private business off the ground.

From a subjective value standpoint, a $1.00 cost could be valued at “close to zero,” and several millions of dollars could be valued at “beyond what I could reasonably expect to earn in a liftime,” that is “nearly infinite.” The life changing nature of the money is what *could* make it potentially worthwhile, even to those who know the odds are approaching “almost impossible to win.”

The odds of winning Megamillions is one in 176 million. Although rumor has it that since Powerball is going to push their odds to 1:195m in January, they may make Megamillions harder to win soon too.

So in theory you would need to be able to win $175m for the jackpot to be ‘fair.’ In other words, the odds you got paid, would be the odds of actually hitting.

Of course, the prizes they advertise in American lotteries are just marketing lies. Because in reality, the government taxes you massively (which they don’t in some other countries like the UK). And also they give you far less if you take a lump sum.

So you’d probably need to see a jackpot of $600m or more before the net prize would be even close to the odds of winning. And even then, it is not unheard of for more than one ticket to win, in which case you”ve just lost your odds again.

The largest ever jackpot won was in March of last year. The advertised prize was $390m, although the real prize was $223m. Two separate tickets won, so each person got $111.5m before tax. So as you can see, so far at least, the jackpot has never been high enough to pay fair odds for the prize, and it probably never will.

At the end of the day, you’ve just got to look at it that someone has to win it sooner or later, and hope for the best. Because in reality the odds are very much against you, with a ‘house edge’ far worse than any casino game. It is a sucker game no matter how you look at it, and there will never be a mathematical justification for playing. It’s just fun, and a donation to charity.

I don’t know if you have to factor in the fact that once the jackpot does get that high, there would also be an increased likelihood that someone else will also have the same numbers you do, which would cut your prize in half.

To answer your question…
The prize would have to be greater than $21,085,384,320.
That’s twenty one billion eighty five million three hundred eighty four thousand three hundred and twenty dollars!!