It's pretty common to encounter a lottery that has a positive expected value.
No, it isn't. You have to include the probability of sharing the prize with someone else which always reduces the expected payout to negative.
Even if a lottery has a negative monetary EV it may have a positive utility EV.
It's basically a transparency thing. Otherwise (well, probably in any case), you'll have people saying the lottery was rigged and the money went to relatives of the the officials running the lottery or something along those lines.
[Oops: That was meant to be a reply to why state lottery recipients sometimes have to be public.]
You're confusing the lottery with the McDonald's Monopoly game