Is buying a lottery ticket ‘effort’?

Why is the expectancy criterion irrelevant to anything when it comes to a single experiment?

Because if a person wants to risk a little money for a lot, there is no reason not to do so. He can succeed. For example, a person who has a hundred shekels and needs a million to live. Isn't it right for him to invest 10 shekels for a 1 in a million chance of making a million? That is his only chance in life.
Alternatively, a person needs to invest a hundred shekels to make a million with a 1 in a thousand chance. The expected profit is positive, but there is still no reason to invest money in an experiment that probably won't succeed. The expected value is irrelevant.
And in general terms, the expected value is the expected result if a lot of experiments are done (the law of large numbers), but in a single experiment any result can come out (even if with a small chance). In such a situation, the expected value is a rather arbitrary criterion.

Hello Rabbi!
Most people I know don't buy a lottery ticket just once, but literally their entire lives, expecting to win in the future, so it's possible that the issue of expectancy exists here. (Although the numbers aren't really large)

In the Rabbi's opinion, is it also necessary to flee to a shelter when there is an outcry in Tel Aviv because of rockets from Gaza for the same reason that the investment is small and the profit is huge, and this is an isolated case even though the chance that the rocket will hit it is zero?

That's still a negligible number.
Regarding going to the shelter, a few things. First, I didn't say anything about whether it makes sense or not. There's no set answer here. Everyone makes their own considerations according to their preferences, and the same goes for going to the shelter. Those who don't go are acting in a completely logical manner. But there's another consideration here, precisely because of the law of large numbers. If I were alone in the country and they announced on the radio that there was a red light, I would continue walking down the street and going about my business as usual. There's no reason to act based on such a small chance of being harmed. But there are millions of citizens in the country, and this small chance could materialize for one of them. Therefore, the categorical order tells us that we should act in a way that we would like to have as a general law, meaning that everyone should act this way. And if the question is about everyone, it's clear that we should go to the shelter. What's more, harming a citizen is a systemic failure of the state and we have a responsibility for all citizens and the state. I dealt with this in my lessons on individual and public in Halacha and in general: https://soundcloud.com/mikyabchannel/sets/y7bf7zt3bcib