Some fear of bears and cougars, or any wild animal for that, is rational. Wild animals are unpredictable. You can mitigate the probability of bad outcomes with smart behavior, that others have cited above.
Statistics from years past may not indicate current probabilities. Particularly with cougars. There has been an uptick in people/cougar encounters as civilization pushes its way more into their territory. Probabilities are still low, however. The number of people killed by deer in the US is not particularly relevant, in my opinion, because there are far more people exposed to deer than backpackers. The specific statistics regarding backpackers are more relevant. To me the number of injuries, not just deaths, need to be considered.
I always check with the land managers (usually when I pick up my permit I ask) regarding any "problem" bears that have been reported in the area that I plan to hike. And if bear spray is recommended, I take it.
I have had numerous bear encounters and they have not in any way been "fun". I hike in the area in northern Yosemite where they relocate the problem bears from Yosemite Valley. One hike I saw one bear per mile! They do NOT all run away, particularly if you are hiking solo or if they have a cub nearby! Most people's fear is overblown, but on the other hand, the casual attitude that all is "fun" or OK is also in error. Read up on animal behavior, stay informed of local conditions, and have a healthy respect for wild animals.