Question: Hard Drive, Backing up & Security

I've searched the archives and can't find an answer--this might be the dumbest question ever.

I replaced the hard drive in 2006 Macbook and before doing so, backed it up.
I've installed the new drive and everything's fine. I destroyed the old drive because I didn't want people to have access to the usual stuff--cc#s, bank accounts, emails etc. You know, all the stuff that you think you deleted but could be recovered with forensic software.

Here's my question--when you transfer/back up data onto a brand new drive, does it also deposit old information you thought you deleted? I.e., are there emails I deleted 3 years ago deeply buried in the backup? Could forensic software retrieve "lost" data from a backup?

Thanks.

That's not my question ... the old hard drive has been destroyed.

I just want to know if the backup will deposit EVERYTHING onto my new hard drive.

The reason I'm asking is that my son is taking this computer on a long trip and if the computer is lost/destroyed/stolen it won't cause either of us to lose any sleep.

Thanks anyways.