[TomCollins]

You are looking for:
P(No Birthday Found Before Me)*P(Someone's Birthday In Line is the same as mine | Unique Birthdays)

Call them 1-f(n) and g(n).

g(n) is easy, its (n-1)/365, where n is your position in line.

f(n) is a little more tricky, in essence, its "a birthday matched before me, or everyone before me is unique, and I match".

So f(n) = f(n-1) + (1-(f(n-1))*g(n).

These can be calculated out.

So the max of (1-f(n))*g(n) occurs when n = 20, so when there are 19 people in front of you.