It doesn't even need to be that desperate. I understand the math, and I play the lottery every once in a while for the same reason that perfectly rational people visit casinos: it's fun and carries a non-zero chance of a payoff that is more life-changing than the money I spend on it. Being stupid is not a requirement.

We spend our money on plenty of things that offer negative monetary returns in exchange for pleasure, comfort, and entertainment. Premium beer, in my case.

Besides, it's not constructive to consider only the straight monetary expectations of lotteries when discussing their socioeconomic implications. Poor people are generally not buying tickets because they think it's the mathematically correct move.

Yes, it can be better than doing nothing, but it's a rational choice only if there isn't anything else you can do that offers a better chance of multiplying your money by 25.

Consider a lottery that sells tickets for $1 and pays back an average of 50c per ticket, with only one prize: the $500k jackpot. (This is pretty much the most favourable case for trying to raise the money by lottery.) Then each ticket has a 1/1000000 probability of winning, and (roughly) if you buy $20k of tickets then you have a 1/50 chance of winning.

Instead, you could go to a casino and bet all your money on one spin of the roulette wheel. Assuming that the game is fair but has single and double zeroes, your chance of winning is 1/38 (instead of 1/50) and if you win you get $720k (instead of $500k).

Or you could sink a lot of money into a troubled company whose stock price is very low and hope it recovers dramatically. I don't know how often such opportunities exist that have even a 1/50 chance of paying off 25x in a hurry, but I doubt it's that rare.