Birthday Problem: Why isn't the probability 253/365

Question

Consider a set of $23$ unrelated people. Because each 23 pair of people shares the same birthday with probability $1/365$, and there are $\binom{23}2 = 253$ pairs, why isn’t the probability that at least two people have the same birthday equal to $253/365$?

Answer 1

Let $A$ be the event that some two people have the same birthday. For $i < j$, let $A_{i,j}$ be the event that persons $i$ and $j$ have the same birthday. Then, $\text{Pr}(A_{i,j}) = \frac{1}{365}$, and your calculation is essentially that $$ \sum_{1 \le i <j \le 23} \text{Pr}(A_{i,j}) = \sum_{i,j} \frac{1}{365} = \frac{\binom{23}{2} }{365} = \frac{253}{365}. $$ But unfortunately, $\text{Pr}(A) \ne \sum_{1 \le i <j \le 23} \text{Pr}(A_{i,j})$, because even though $A = \bigcup_{i,j} A_{i,j}$, the events $A_{i,j}$ are NOT disjoint: it could be that multiple pairs of people have the same birthday.

On the other hand, the total number of pairs sharing a birthday is $1$ birthday for each $A_{i,j}$ that occurs; therefore the expected number of pairs sharing a birthday is exactly what you have calculated: $\sum_{1 \le i <j \le 23} \text{Pr}(A_{i,j}) = \frac{253}{365}$.

Answer 2

explanation for prob 1/365 we consider one pair (a,b) of people...the sample space for a particular pair has 365*365 elements...and out of those the elemnts which have same birthday dates will be 365...so thats how the prob is 1/365....

and considering that prob for one pair is 1/365 and there are total 253 pair then there will be 253/365...is wrong beacause they are not mutually exclusive....

according to me the correct answer can be like this.. prob atleast one two people have same bdy=prob(2peoplesame)+prob(3people same)+....prob(23 people same)

=1/365+1/(365)^2 + 1/(365)^3+...1/(365)^22 =1/364(approx)

Answer 3

253/365 is the expected number of pairs with the same birthday. However, you might have two or more pairs. Your formula would count that as 2, while the answer to “anybody with same birthdays” is “yes” and not “yes yes”. So your number is too high.