Final answer:
The most likely solution to a Time Machine backup volume being read-only is to change its permissions. If that doesn't work, checking Time Machine preferences or formatting the volume might help, but buying a new drive should be a last resort.
Step-by-step explanation:
The question, 'My Time Machine is unable to complete a backup. When it attempts to back up, an alert message appears that states, 'The backup volume is read-only.', suggests an issue with the permissions or the formatting of the backup volume used by Time Machine, which is a backup software application designed for Mac computers. Among the options provided, B) Change the backup volume permissions is most likely the correct approach to attempt to resolve this problem. This involves checking the permissions and ensuring that the volume is not set to 'read-only' mode, which prevents any new data from being written to it.
If changing permissions does not resolve the issue, then other options such as A) Format the backup volume or D) Check Time Machine preferences may be considered. Formatting should be a last resort as it will erase all data on the drive. As for purchasing a new external drive (C) Buy a new external drive, it is unnecessary unless the current drive is failing or presents hardware issues after other troubleshooting steps have failed.