We are evolutionarily prepared to fear snakes and lions, even with they are caged and harmless, even though this seems irrational. Yet we rationally eat burgers and hotdogs even though we know they are harmful. People fear being one of the 48 average annual airline fatalities while they think nothing about becoming one of the more than 30,000 annual vehicle fatalities. Perception controls our actions more than reality. This gap between our fears and the facts is called the perception gap.
A tool, called the psychomettic paradigm, describes the little tricks our brain uses in evaluating risks. Optimism bias gives a rosier view of the future than the facts might suggest. Confirmation bias leads us to prefer information that supports our thoughts and discounts information that is contrary to those thoughts. We also tend to conform out thoughts to those of groups with which we identify ourselves. However, it is heuristics, the subtle mental stages that give rise to such biases, that most affect our risk perception.
The availability heuristic says that the easier a scenario is to conjure, the more common it must be. It is easier to imagine a tornado destroying a town than it is to imagine our arteries being clogged from too much saturated fat in our diet; therefore, we tend to think tornados are more common and more of a threat than heart disease. The representative heuristic makes us think something is more probable if it part of a set known characteristics. That guy up ahead is wearing baggy pant, gold chains, and backward ball cap, therefore it is probable that he is gangsta. However, the more influential heuristic is the affect heuristics. It is that little voice that creeps into our decisions. The positive feelings associated with something tends to make us think it is the more beneficial, while negative correlations make us thinks something is riskier. People start smoking even though they know about all the dangers because they think about the pleasure they get from it rather than the risk of dying from it. Immediate pleasure wins over the chance of dying.
Therefore, if your sole purpose of training in the martial arts is for self-defense, then your time could better be spent on other pursuits. Basic self-defense techniques may be learned rather easily but to become proficient and stay proficient at them takes a lot of time, money, and effort and your chances of ever needing the skills are slim. If you are training in the martial arts for other purposes, such as fitness, self-confidence, self-esteem, sport, hobby, or just for fun, then you are not so concerned about the time, money, and effort it takes to be proficient, and self-defense is a useful side benefit.