Why does probability-neglect happen? It's partly a psychological thing; imagining something happening and how we would feel about it is easy, natural, intuitive, but thinking about probability is difficult, mathematical, unfamiliar. And it's also partly a media problem; we have much more news reporting now than we did, say, thirty years ago. There were no 24-hour news channels then, only a small number of TV channels, and no internet. The news machine is voracious, and so when there is an accident or a disaster or any sort of human tragedy, it is reported and analysed endlessly. This makes us think that events which are actually very rare happen frequently, and perversely, events which are relatively common are under-reported precisely because they're not news.
Quiz: What's the safest way to travel? How much safer do you think it is to travel by car than to walk? A bit? A lot? Is a train safer than a plane? The best numbers I could find were fatalities per billion passenger kilometres for 1999:
Mode of Travel
Deaths Per Billion Passenger Km
Motor cycle 112