How to check whether a background job is alive? (bash)
You can get the PID of the most recent background job with $!. You could then check the exit status of ps to determine if that particular PID is still in the process list. For example:
sleep 30 &if ps -p $! >&-; then wait $!else jobsfi
You can check if a signal is deliverable
./script2 &myPid=$!sleep 5if kill -0 "$myPid"; then script3 & waitfi
To expand on fduff's answer you can use the jobs builtin:
if jobs %%; then exec ./script3.sh & waitfijobs %% prints the job ID of the most recent background process (the "current job") and returns 0 on success or 1 if there is no such job.