Get yourself a big walking stick

We haven't had a big cat sighting in town for about 3 or 4 years now, but with the urban growth of the area, & the ban on hound hunting, the cougar population mushroomed (which accounted for them moving into the outskirts of town).
At one point we had one living on Sehome Hill, which is a wooded green space surrounded by a highly urban neighborhood, the college, & a high school!

It seemed they lost their fear of man & dogs around here, and at a popular lake & park about 7 years ago, two women on a nature walk were attacked by a big cat. The only thing that kept it from doing serious damage was one woman's walking staff keeping it just out of reach! When my daughter's grade school class had a field trip there, I'd go along & walk the trail in the back of the pack, watching both for stragglers and the brush & trees. Better to be safe than sorry

The deer population in the area has mushroomed too (real road hazards), so this might account for the decline in local cougar sightings (they're not hunting further out for chow). I don't know...

Anyhow, unless you have the factors we had here, you'll probably be alright. Usually big cats are very solitary & really don't like human contact.
I remember as a kid (about 11 or 12 yrs old) while spending the summer at my grandfather's farm in the SW part of this state, we were going up into the woods to check the spring level where the communities water was piped in from. While trudging up the hillside, we had a big cat (close but unseen) "scream" at us....liked to scare me half to death (and just as the old tales say, it did sound like a woman's scream). My grandfather, a man of the woods (a life long hunter & logger, both in the Ozarks & here in Washington state), paused, said "its just a cougar, probably warning us off from a fresh kill", and he just continued on his way.
We never did see that big cat, nor were troubled by it on the way back down.
I'll never forget that close encounter.

(Sorry for the long winded response, I got on a roll )