Will a hard copy of the DLCs work on a dital copy Of Fallout New Vegas?

I just bought fallout New Vegas for 2$.95 on Xbox game store

And realize I forgot the DLCs but it's 25 to 30 bucks for the DLCs by themselves so I thought about getting a hard copy for 9.99 with the DLCs, I prefer digital in case I break my disc or lose it but will the hard copy of the DLCs work on the digital copy of the game?

If so will I even need the disc after I install the DLCs?

:)