Chances of winning a raffle when winning tickets are returned to bucket each time.

first time posting here - I like the site. I have a raffle odds question.

I'm doing a raffle where I give away 365 prizes. Every winning ticket is returned to the barrel after each drawing (and info it recorded) before drawing for the next prize. There are 9,999 tickets sold on this raffle. What are the odds of winning if someone were to purchase just 1 ticket?

thank you!

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So far, all I've figured is since there is 365 prizes and 9,999 tickets - 9,999/365 = 27.4 or 1:27 - am I way off? – Duck Dec 10 '13 at 7:00
Note that the probability $p$ of winning at least once is $1-q$, where $q$ is the is the probability of not winning at all. To not win at all, you must lose on every draw, and each draw doesn't depend on the other draws. Can you see how this leads to a value for $q$? – Sam DeHority Dec 10 '13 at 7:44
Thank you Sam. I'm actually not doing a math problem - just need this figured out for a raffle I am literally holding. I need the answer and evidence of how it's found. – Duck Dec 10 '13 at 15:26

Hint: what is the chance that the ticket loses the first draw? To not win a prize, the ticket needs to lose $365$ draws in a row. The chance of winning is one minus this.
An approximate chance of winning is to compute the chance of winning on the first draw, then multiply by $365$. This gives too large a value (try it!) because this is the expected number of prizes won, but sometimes you will win more than one. The chance of winning at least one is then less.