Wealth is could be earned by luck but it's not applicable to all people, so you would really need to earn it by working hard, being consistent, being knowledgeable and disciplined. So let's say that $50 dollar won't be enough for you to become wealthy to invest are you just gonna wait for your luck to achieve? People would literally take extra jobs or part time if you're a student just to have enough funds to invest. Well those examples you've mentioned might be needing a huge amount but little business would acttually a good start cause most successful person achieve it with experience. Luck and opportunity won't be siding on you if you can't even make an action to attain financial freedom.
When you say luck, I think of inheritance from a wealthy father or a family that has been wealthy for decades. With the state of the economy we are all in, it is very difficult to get lucky with wealth, but you can win the lottery or a gift, and these days no one is gifting anyone with money. If you are in this situation, the best thing to do is to get a job and gradually build your wealth. And since the world is changing now its better to even work smart than work hard in some circumstances, our attention to be channeled to what aspect have high demand. Small beginning are appropriated. But I feel 50 dollar is small to start up a good business currently. Either small or big management matters too.