Would it be better to develop my own facebook app or outsource it?

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Well it depends.

Do you have any experience with programming or development? If so, then there is no reason not to attempt developing this app yourself because you could probably learn the relevant skills quite quickly. If not, it would be better to outsource it. You could try to develop it yourself, but with no past experience, it would take so long that you might find someone else beats you to it. Or your finished product might not end up anything like how you wanted it.

"will I have full ownership over the app and the money or will the person who developed it for me be in on it?"

That's something to negotiate with whoever you outsource the development work to. Generally, you can pay them a fixed rate for the development work and then that's it, you own the app. But there are people who expect royalties. You really have to check with each developer and see what there terms are.