George E.P. Box, a statistician known for his body of work in time series analysis and Bayesian inference (and his quotes), recounts how he became a statistician while trying to solve actual problems. He was a 19-year-old college student studying chemistry. Instead of finishing, he joined the army, fed up with what the British government was doing to stop Hitler.
Before I could actually do any of that I was moved to a highly secret experimental station in the south of England. At the time they were bombing London every night and our job was to help to find out what to do if, one night, they used poisonous gas.
Some of England's best scientists were there. There were a lot of experiments with small animals, I was a lab assistant making biochemical determinations, my boss was a professor of physiology dressed up as a colonel, and I was dressed up as a staff sergeant.
The results I was getting were very variable and I told my colonel that what we really needed was a statistician.
He said "we can't get one, what do you know about it?" I said "Nothing, I once tried to read a book about it by someone called R. A. Fisher but I didn't understand it". He said "You've read the book so you better do it", so I said, "Yes sir".
Box eventually worked with Fischer, studied under E. S. Pearson in college after his discharge from the army, and started the Statistical Techniques Research Group at Princeton on the insistence of one John Tukey.