If you're kind of poor (not starving though), it makes sense to play the lottery. $10 a month isn't going to cause you to starve, but there is a possibility you'll get a decent chunk of money (not jackpot perhaps, but enough to substantially improve your standing say). Whereas for someone who is well off, getting a few thousand dollars isn't such a big deal.
The usual reason for disagreeing with what you just said is that for most people the utility gain from (say) a 1/6000 probability of getting $3000 is less, not more, than the utility loss of certainly spending $1, and it only seems otherwise because we tend to round small utility differences to 0 mentally. I think this is correct. But here's another reason why you're wrong.
First of all, let me correct something in the previous paragraph. Of course hardly anyone plays the lottery just once, so let's consider playing 10 times per month in a lottery whose tickets cost $1 and whose only prize is $3000, with probability 1/6000. (Oversimplification, of course; I'm going with basically the figures you mentioned, and a fairly typical 50% take for the people running the lottery). So, if you do that for a year then roughly speaking you are paying $120 for a 1/50 chance of getting $3000.
So, suppose there's some person for whom the utilities really do work out the way you describe: getting $3000 with probability 1/50 is worth more than $120. And suppose that person could instead invest their money somewhere reasonably safe and get a return of 1% per annum over (inflation plus their temporal discount rate). For simplicity of exposition, I shall ignore inflation and discounting in what follows, so that's just an annual return of 1%.
Then instead of spending $120 on the lottery over a year, they could invest ~ $119.40 over the same period, at the end of which they will have $120 which they can put into the lottery. Same result, less money spent.
So. Unless your temporal discount rate is higher than the best rate of interest over inflation that you can get, playing the lottery now cannot be your best option -- at least, not if your motivation is that you think the utility gain from having more money when you win is worth the utility loss from having less money because you spent it.
In less-crappy economic times than we've had recently, you can typically get a rate of return that's quite a bit better than inflation. In which case, unless your discount rate is really high, you should not play the lottery with the intention of increasing your net utility.
(Of course, if your discount rate is really high, it follows that you should never save anything: anything you would save you should spend immediately on things that give you immediate pleasure, or put into the lottery at once.)
If you're poor, buy one lottery ticket where the draw is as far into the future as possible, or a scratch lottery ticket and don't scratch it until it's about to expire. As long as it is not decided, you are potentially a winner.
Except that they lack the utility of playing the lottery for an entire year, which is already worth more than $120 to them, as we can see from the fact that they play the lottery in the first place. :)
The person I was replying to wasn't making that argument, but a completely different one: that the utility from the potential winnings outweigh the costs for some poorer people. I think that's monstrously implausible.
You could say this is irrational, or you could just say it's the way people are.
"Investing" small amounts to achieve a big pay-off also hits the problem that large windfalls aren't psychologically or socially very beneficial.
That said, it's deplorable government is running these programs. Only 40 years ago this was the illegal domain of mobsters and sleazeballs. Now the state has muscled in on the action is legitimizing the moral message that you can get something for nothing.