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도와주세요. 질문이 있어요.

든 in your sentence is a short form of ~ 든지, which translates to ~ever (as in whoever, whatever, etc.)

Here are some examples:
누구든지 = 누구든 = whoever
뭐든지 = 뭐든 = whatever
언제든지 = 언제든 = whenever

Here are a couple sentences using the above example words:
누구든지 여기서 먹을 수 있어 = 누구든 여기서 먹을 수 있어.
Direct translation: Whoever can eat here.
Anyone can eat here.
너는 뭐든지 할 수 있어 = 너는 뭐든 할 수 있어.
Direct translation: You can do whatever (you set your mind to).
You can do anything.
언제든지 들르세요 = 언제든 들르세요.
Direct translation: Please come by whenever (you want/can).
Please come by anytime.

So, the translation of your sentence, "복권에 당첨될 수 있는 가능성이 누구에게든 있는데 왜 안 사?" would be like this:
Direct translation: Whoever has the possibility of winning the lottery, but why don't you buy (lottery tickets)?
Anyone has the possibility of winning the lottery, so why don't you buy lottery tickets?

가능성이 있다: There is a possibility = there is a chance.
나에게 가능성이 있다: I have the possibility = I have the chance.

FYI, the link to the similar grammar is: http://www.koreangrammaticalforms.com/entry.php?eid=0000000906

Hope this helps. Please let me know if you have any questions.