Intuition about the birthday problem

In the birthday problem, where we are trying to find the probability that another student in the class shares our birthday. I understand that we can use the complement rule to find the probability that a student does not share the same birthday as us.

P(S) = 1- P(S*) = 1 - (P(S1) * P(S2))

Can’t we just find the probability of having the same birthday as us and add up to the number of students i.e

P(S) = P(S1) + P(S2)

Is there anything wrong with this approach?

For example with 2 students
Complement method
1 - ((364/365) * (364/365)) = 0.005471945956089

Direct method
1/365 + 1/365 = 2/365 = 0.005479452054795

With 100 students
Complement method = 0.2399329261841
Direct method = 0.273972602739726

The answers are different.

Please can someone clarify what I am missing here?

Your second method gives the probability that any pairs of two people in the class would have the same birthday.

It lacks the constraint that one of the birthdays specifically be your birthday. That’s why its probability is higher.