Question

The birthday paradox concerns the probability that, in a set of randomly chosen people, some pair of them will have the same birthday. By the pigeonhole principle, the probability reaches 100% when the number of people reaches 367 (since there are 366 possible birthdays, including February 29). However, 99% probability is reached with just 57 people, and 50% probability with 23 people. These conclusions include the assumption that each day of the year (except February 29) is equally probable for a birthday.

What is your response?

Answer 1

**and why

Answer 2

@jhonyy9

Answer 3

The law of combinations says: the number of combinations of interacting elements increases exponentially with the number of elements. In the question, it asks the probability that ANY two people have the same birthday as EACH OTHER.
The number of pairs of n people is given by:
\[\left(\begin{matrix}n \\ 2\end{matrix}\right)=\frac{n\times(n-1)}{2}\]
Obviously the number of pairs increases rapidly as n gets larger. When n = 23 there are 253 possible pairs of people. Looking at the probability that none of the 23 people share the same birthday, we can reason as follows:
For two people, the probability that the second person has a different birthday from the first is 364/365. Then the probability that those two are different AND that a third person has a different birthday from either of them is 364/365 * 363/365. Continuing this reasoning, the probability that none of the 23 people share the same birthday is given by:
\[\frac{364}{365}\times\frac{363}{365}\times\frac{362}{365}\ ............\ \frac{343}{365}=0.49\]
The probability that some of the 23 people share the same birthday is therefore:
1 - 0.49 = 0.51

This explanation will (hopefully) help you to fully answer the question.

Answer 4

@kropot72

What about the exclusion of February 29th and the 1/4 day (leap year)?

Answer 5

I believe that the question excludes the consideration of leap years by effectively saying that February 29 has zero probability.