Birthday Paradoxon

Question

So I saw this interesting problem: https://en.wikipedia.org/wiki/Birthday_problem

And I am not the best at probability, so my question is why I cant calculate the probability with

P (2 in n same birthday) = 1/365 * 2/365 * ... * n-1/365

and have to use this instead?

P (2 in n same birthday) = 1 − P (2 in n not same birthday)

I understand how it works, my problem is that this would not be my first approach on this problem.

Answer 1

Let's see how your formula works with small, easily calculated numbers. What is the probability that two people out of $n$ share the same birth season? You claim it would be $$\frac 14 \frac 24 \frac 34 \cdots \frac {n-1}4$$ So

• a group of two would have $\frac 14$ chance of sharing a birthseason (correct!)
• a group of 3 would have a $\frac 14\frac 24 = \frac 18$ chance of sharing a birthseason (adding another person made the probability go down??)
• a group of 4 would have a $\frac 3{32}$ chance of sharing a birthseason (???)
• a group of 5 would also have only a $\frac 3{32}$ chance of sharing a birthseason!

Which is ridiculous because for $n > 4$ the real probability is obviously always $100\%$.

I'm not sure why you thought this was a valid scheme for calculating the probability, but it is not even close.