By moving a file to Trash and then emptying the trash, or by doing a quick format of a traditional hard drive (i.e. not a solid state drive), you're actually not deleting files. Instead, all you are doing is deleting the information about those files. This means the operating system has absolutely no idea those files exist, let alone where on the drive they exist.
Okay, let me explain this by using an analogy.
Imagine you’re at a library and this particular library contains 100,000 books. All these books are indexed in the library’s catalog. This catalogue is accessible via a computer that allows you to search by title, author, date, etc. Most importantly, each index tells you exactly where that book is located. This makes it easy to find what you’re looking for when you need to. You simply conduct a search of the catalog and it’ll tell you exactly where that book is (i.e. what section, row, and shelf it’s sitting in).
One day, someone accidentally deletes the record of a particular book from the library’s catalogue. The book itself is still there, sitting in exactly the same spot. But no-one knows it’s there and/or where it is!
The next day, someone breaks in and steals the computer containing the library’s catalogue. All of the books are still there, but if you walked in and wanted to find a particular book about Steve Jobs, you’d have no idea if that book is in the library and, if so, where to find it!
Now imagine that your hard drive is that library. It contains 100,000 files (documents, photos, videos, music, etc).
When you move a file to the Trash and then empty it, it’s akin to someone accidentally deleting the record of a particular book from the library’s catalog.
When you erase a hard drive, that’s akin to someone stealing the library’s entire catalog.
So, using our analogy, the books are still on the shelves. They haven’t gone anywhere! But if you wanted to find that one book you’re looking for, well, you’d have absolutely no idea where to look.
Likewise when deleting files or erasing a hard drive. Your computer has no idea whatsoever what data is on that hard, or where it is. So, as a user, if you navigate around the hard drive you won’t see those files because your Mac isn’t aware of them and therefore isn’t displaying them. They effectively don’t exist.
But, if you were to use data recovery software, then that’s akin to a library hiring someone to go in and walk up and down every row of bookshelves and opening each and every book and taking notes of their titles, authors, location, etc. The data recovery software has no idea what’s on your hard drive initially, but it will interrogate every block of space to actually see what’s there and can then recover it. In other words, it’s making what’s already there visible again.
When you empty the trash or erase a hard drive and then start saving new files to your computer, over time macOS will just overwrite the previous data because it just doesn’t know there’s any data there. It just sees it as free space and therefore available to write new data over. That’s why recovering data is always more successful soon after you’ve deleted something, because there’s less chance it’s already been overwritten.
If you want to be fairly certain those intimate photos aren’t recoverable, you can:
- Secure erase the hard drive (since you’re on El Capitan, read this question and its answers How to get the "securely erase" function of Disk Utility on El Capitan & Sierra)
- Manually fill the empty space of the hard drive with other data, preferably large files (e.g. a whole heap videos) and then when it’s full, delete this files. In this way, the blocks containing the previous photos should be overwritten with other data.
IMPORTANT: If your daughter has Time Machine backups, then all those intimate photos will almost certainly be in those backups as well.
A final word...
Another option users can take is to encrypt their Mac startup drive. This will secure your data because everything on the disk is encrypted. If you delete files, these will be unrecoverable or, more to the point, they can be recovered, but they're encrypted and therefore inaccessible to anyone who doesn't have the right credentials (e.g. login password, recovery key).
For more info on how to use FileVault refer to: Use FileVault to encrypt the startup disk on your Mac.