Profit simply means that an activity is worth it: that the result will be worth more than the costs to reach the result. Profit is the aim not just of corporations, but also of government.
For example, a government should not build a road unless the people using the road will collectively save more in transportation costs than the costs to build the road.
The problem is that for many projects, the net benefit is unknown at the start. In the road example, before a road is built, it may not be clear how much it will cost to build a road nor how much people will save from using the road. The best we can do is try to estimate.
But just because we have to estimate doesn't mean that we ignore profit. Before a project is undertaken, we should still try to figure out if its worth doing. We just can't be sure.
When we say that government funds "risky" projects, this can mean that the projects have a high probability of failure but are still worth funding because the probabilistic expected payout is positive, or that the projects are really not worth funding because we really expect that there will be no net benefit. If we're talking about the first, then the private sector will fund it without government. For example, YCombinator routinely funds companies that have a high probability of failure. But overall it expects a positive payout. So startups get funded.
But if we're talking talking about the second type of risky, as in no expected return, then really no one should be funding these projects. Why should the taxpayers take risks collectively that no one is willing to take individually?
It doesn't make sense for an individual to try to fund retirement by buying lottery tickets, no matter how much money they spend. Therefore, government should not be trying to fund government pensions by buying lottery tickets.
In practice, governments seldom openly advocate investing in projects with negative returns like a lottery. They propose funding projects that the private sector won't fund. But even if a project is expected to be profitable, it still should not be funded at the expense of projects with higher expected profit. And because we have limited resources, the funding of low-expected-profit projects means that other higher-profit projects can't be funded.
Again, I refer you to Henry Hazlitt's Economics in One Lesson, chapter 6 this time: http://steshaw.org/economics-in-one-lesson/chap06p1.html.