Many people have heard of the birthday problem: "How many people do you have to get in the same room so that there is a 50-50 chance that two of those individuals share the same birthday?"
The answer is, perhaps, surprisingly few: 23. Why so few? Many people misinterpret the problem as "How many people do you have to get in a room so that there is a 50-50 chance that one of them will have their birthday on March 5th?" In this case, the answer is 183. But this is for a specific day, not just any shared birthday.
Suppose that there are 10 people in a room, none of whom share a birthday. For the eleventh person showing up, that person cannot have the same birthday as any of the ten who were already there, so 355/365.
Why is this the
On Skeptic's Guide to the Universe, the presenters once announced how few people were needed to
If you work out the mathematics, the likilihood of two individuals sharing the same birthday with elven or fewer persons is one in four.
Now, what is the likelihood of two people sharing the same birthday? One in 365. What is the likelihood of someone having their birthday on March 5th? One in 365. The probability of two random events coinciding equals the probability of one random event equaling a fixed value.
Thus, here is an alternative birthday problem, where one can privately verify the result, while not giving away too much information.
Ask the first person, is your birthday before January 1st? Of course, the answer is "no."
Ask the second person, is your birthday before January 2nd? There is now a one-in-365 chance of this occurring.
Ask the third person, is your birthday before January 3rd? Now there is a two-in-365 chance of this occurring.
Repeat this, always asking the next person has a birthday on before the next day.
Proceed as follows: Only allow those audience members to participate if they are willing to give proof of their birthday if necessary. If a member of the audience refuses, just pass over that idividual. Then keep asking the next person, and the person thereafter, until
What is nice about this system is that, except in the very unlikely situation that the second person's birthday is actually on January 1st, there is an ambiguity as to when the person's