I think taking a risk always implies the potential for something negative to occur. If there is a positive outcome possibility, I tend not to use the word 'risk'; For a lottery, I would use the phrase 'taking a chance' - it's more neutral and includes both positive and negative outcomes. For example, one would take a chance on winning the lottery (the negative possibility is the risk of losing money that is outweighed by the benefit - the hope that one might win, and the possibility of actually winning). Maybe the usage is different in other languages and cultures or to other people.
Some risks are voluntary (e.g., rock climbing) and some are involuntary (travel by airplane), although one could argue that all risks are to some degree, voluntary (e.g., I could move away from my city - but I choose to live in my city which is near a major subduction zone even though there is a risk of a large earthquake damaging my property or harming me.
Maybe people expose themselves to risk because they enjoy benefits from the activity and have decided that the risk imposed is tolerable. I used to rock climb because I loved the outdoors, the friendships, and the views from the top of the climb. My friend chose not to climb because those benefits did not make up for the risk of falling.