Re: what are the chances of two people being born at exactly the same time?

According to the Population Reference Bureau, the annual birth rate for the entire world is about 133 million births per year (22 births per 1000 individuals per year times a world population of 6.067 billion individuals). Since delivery can be a lengthy process, I would assume that an accurate interpretation of "at exactly the same time" would be to the closest minute. Otherwise, we would need to define a specific occurence in delivery - clearing the canal, first breath, cord cut, etc. - as our reference point to define the actual second of birth. With that assumption, there are 525,600 minutes in a year, which would give a world birth rate of around 254 babies per minute; so the chance of two people being born within a minute of each other is nearly absolute (within 60 seconds in a minute, you can see that the world birth rate is over 4 births per second).

In case you weren't interested in the world population, here's the calculations for the United States (again using the data from the Population Reference Bureau). There are just under 4 million births per year in the U.S., which comes to 7.6 births per minute: about 1 birth every 7.8 seconds. That's still high enough to say that the chances of two people in the United States being born within a minute of each other is very high. If you were referring to their being born at the same time in the same hospital (or city), then you would have to refer to the birth rate of the local region of interest.

If you're question was more along the lines of, "what are the chances of meeting someone who was born at the same time as you," then the answer requires more mathematics. Assuming an even distribution of births across the year (this is not the case, but the statistics get messy otherwise), the chance of a person being born at a given time is 1/525,600. Since your birth time is fixed, the chance of someone else having the same birthday and time is 1/525,600 or 0.0002% - Note: this is not being born at the same time, but having the same birth time, i.e. 7:34pm June 8th of any year. On the other hand, the chances of anyone in the U.S having been born at the same time on the same date as you would be 6.6 (other people born at the same time) divided by 275,600,000 (the population of the U.S.), which comes to about 1/41,800,000 or about .0000025%. In fact, harking back to the "Birthday Problem", you would need a room of over 33 people to have even a 0.1% (1 in a thousand) chance of any two of them having the same "birth time", ignoring their years of birth. (My calculator, fingers, and patience prohibited me from working out the exact number of people needed to have this same likelihood of two people being born at exactly the same time meeting in the same room.) So, while the chance of two people being born at the same time is very high, the chance of their meeting each other is extremely low.