While planning the wedding a bride wants that her wedding date not be held in any month where a guest or wedding member has a birthday (herself and the groom included). For example: if the groom's birthday is in January then that is no longer an eligible month for the wedding to be held in.
At what total wedding size ($n$ = number of people in the wedding) does the probability become $0.5$ that the wedding will be cancelled (due to lack of potential months within which to set the date)?
Generalized marble problem: I have a bag with $k$ marbles, each a distinct color. I then ask $n$ people to pick a marble and note its color before putting it back. How big can $n$ get before the probability of all $k$ colors being picked becomes $.50$.
Wintermute, your problem is an instance of the classic coupon-collector problem represented by the possible statement:
Notice that in you wedding party problem, guests represent boxes of cereal, and the coupons represent the month of birthday of each guest. That is, you have 12 different coupons, one for each month of the year. An easier version might assume, as crude approximation, that birthdays are equally likelly in each month (1/12 probability).
There are lots of references about how to answer several questions about this problem. Some are easy to answer, others are difficult. Have a look at the Wikipedia reference for the Coupon Collector Problem for more details and a starting point for your research.
Let's start with a simpler question: suppose you have 12 guests and want the probability of having no party, as each one has birthday in a different month:
$$\text{Pr(no party with 12 guests)}=\dfrac{12!}{12^{12}}\approx 5\times 10^{-5}.$$
And if $N$ is the number of guests required to the no-party situation, it is true that $E(N)\approx 37$.
Now you can take from here.