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(S*1) * P(S*2))

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?