I'm currently reading a book called "Death in Big Bend" by Laurence Parent. I've been hiking this park since the 70's, about 50 trips now. According to Parent's statistics, the most dangerous animals in the park (besides humans) are deer. They cause the most traffic accidents. There have been no bear attacks. There have been a couple cougar attacks that resulted in minor injuries, from sick cats. I've never been lucky enough to see a cat in the park.
My wife and I have had 5 black bear encounters, in Texas and Canada, all which turned out to be fun. Black bear usually run like heck if they see you. If they are human hibituated, they go about their business...you just walk on. They can wreck your camp looking for improperly stored food, but generally you are lucky to see one.
Grizzly bears are a different story but still, deaths are rare.
Read up on your predators. Turn your fear into respect and hike on.