The Crowded Calendar

A host gathers a group of people in one room and wonders whether any two of them celebrate their birthdays on the same date. Assume there are 365 possible birthdays, each date is equally likely, birthdays are independent, and February 29 is ignored. We are interested only in whether at least one matching pair exists; several matches or three people sharing a date still count as the same event. What is the smallest number of people the host needs so that the probability of at least one shared birthday is greater than 50 percent?