I take the techno-elite bus to work (you know, those ones that are becoming increasingly popular with the locals). Since the bus only leaves / departs once per day, I occasionally take Caltrain for one leg of the journey.
Caltrain is a proof-of-payment system. You can simply waltz onto the train without paying, but the conductor issues stern warnings that Caltrain retains the right to check your ticket and issue fines.
Should you actually buy a ticket? To the economically trained, this becomes an intriguing empirical question, solved by the expected-value equation.
Using my Clipper stored-valued card, I ride three zones, which costs $6.75 (see full fare chart). Getting an estimate on the fine is a little tougher. A Yelp post from 2007 says the fine is $300, while another blogger who asked himself this same question in 2012 cites $250. Let's go with the $250 number, since it's more recent, and it can bias my answer toward "you should not pay" (the outcome that all economists are secretly rooting for).
The equation would be as follows: 250x = 6.75, where x is the percentage chance that they'll check my ticket. Turns out that if tickets are checked any more frequently than 2.7% of the time, you should buy a ticket. Things actually are a bit worse than that, as this equation doesn't factor in the shame you may feel if the conductor yells at you, the anxiety you may feel worrying about whether tickets are checked, or the hassle of physically paying the fine. On the flip side, maybe you could contest the assumption that you'll have to pay 100% of the time when they find you without a ticket, depending upon how good you at sweet-talking or claiming to be a clueless tourist. My wife's cousin also suggests that the "the machine wasn't working for me" excuse might work better if you actually have a Clipper card in your pocket instead of arguing that you were trying to buy a paper ticket but failed.
How often does Caltrain check tickets? Estimates vary widely: I have ridden Caltrain from Millbrae to San Jose or vice versa 10 times and have yet to be checked. My wife's cousin estimates that tickets are checked 1 in 6 times. The aforementioned blogger experienced checks on 45% of his rides in 2012.
Even though I've yet to see a Caltrain ticket check myself, I have to put my Bayesian reasoning to work here and assume the true rate is at least 2.7%. So while I can't squeeze any gains out of a government agency this time, at least I had fun trying.