How many days in the year?

Here's a brain teaser: your helpful lab assistant has rounded up a sample of N individuals. One by one, they tell you their birthday. What value of N is needed to determine there are 366 days in the (leap) year? (Or, if you hate leap years, that there are 365 days in the year.)

Variant A: You only know the existence of a day when someone tells you their birthday. So, if the first person says they were born January 2nd, you cannot infer January 1st must exist because "1 < 2".

Variant B: You can infer from January 1st the existence of January 2nd.

Possible acceptable answers include but are not limited to: a probability distribution for getting the correct number (as a function of N), the N which maximizes the likelihood of getting the correct number of days in a year, or the expected value for N.

Variant C: What other ways are there to determine N?

I may post a solution next week to this (or the week after).