How can I check if the external drive is included in the back-up?
Open System Preferences (gear icon on the dock)
Open Time Machine
Click on Options
Here you will see a list of devices and files that have been excluded from backups.
In other words, if you want an attached hard drive to be backed up along with your Mac, the name of the drive should not appear here. If it does, it is excluded, and it is not being backed up.
Click to highlight the drive, and then click the - (minus) button at the bottom of the list to remove it from the list of excluded items. The drive will be backed up on the next pass that Time Machine makes.
If a drive is grayed out in this area, you cannot remove it from the list of excluded items since it is not formatted correctly to work with Time Machine.