Let's say the probability of winning is (so ). The probability to win (at least) once over attemps is equal to the probability not to lose every of the attemps.
Losing times in a row has probability , hence the probability to win at least once in attemps is .
For instance, if you want to make this probability larger than 50%, you need to have . If is small, this is very close to (and not as you wrote). In your case, instead of 13,983,816 / 2 = 6,991,908, it is rather 9,692,843 times someone needs to play lottery to have 1 chance over 2 to be a winner. Nearly 14 lifetimes are necessary for that.
About your last question, someone who (foolishly) would try every combination would have a probability of , that is 63%, to have won at least once.
This comparaison with lifetime is quite convincing, by the way...