Just for the nervous flyers; over the last 20 years; flying regularly in VERY busy airspaces, including the London FIR, Paris and New York, I have only ever had a handful of TCAS TAs, (Traffic Advisories, i.e. warnings of potentially conflicting traffic, but no recommendation to change flight path), but NEVER any RAs, (Resolution Advisories, i.e. recommending action to be taken to avoid actual conflicting traffic).
We do get them in every 6 month Simulator check, to practice the avoidance manoeuvre(s), but I have never had one for real.
Air traffic control in 'civilised' places is very good and safe
PS, Most TAs are caused by your, or other, aircraft climbing or descending more rapidly than 'normal' but nevertheless to a safe flight level above or below the conflicting traffic; so there is no collision risk. Such TAs are easily and quickly resolved by simply reducing their (or your) vertical speed.
TCAS is an extra safety layer on top of the many other tried and tested ATC procedures and systems.