The problem with monopolies

An automobile tycoon has built another factory and needs to populate it with workers. Due to the poor levels of employment, he has a huge pile of applications on his desk. To save himself time, he begins by throwing away the applications of all those born in leap years. Each worker he chooses will take one and only one day off each year - his birthday. So, unaware of the dates of any of the applicants' birthdays, the tycoon begins admitting applicants on a first come first served basis until the admission of any further applicant would cause the probability that at least some 2 workers had the same birthday to be greater than $0.5$ . How many workers will the tycoon employ? $\textbf{Details and Assumptions}$ The tycoon will not do any labour himself so his birthday is irrelevant.