During lunch today, the topic of the "Birthday Paradox" came up while discussing gambling and probabilities. I remembering reading or hearing that there was a high probability that two people would have the same birthday in a classroom of 35 students. After a little searching on Google, I found some pages that basically support what I remember.
The birthday paradox states that if there are 23 people in a room then there is roughly a 50/50 chance that at least two of them have the same birthday. For around 60 or more people the probability is greater than 99%. This is not a paradox in the sense of it leading to a logical contradiction; it is a paradox in the sense that it is a mathematical truth that contradicts common intuition.
Some Additional Links below: