Birthday Problem Alteration

The well-known birthday problem asks how many people must be in a room before the probability of two or more people people sharing a (any) birthday is $>0.5$ and the answer is 23; a common alteration to this is how many people must be in a room before the chance of at least one person sharing your birthday (i.e. one specific date) is again greater than half, the answer is 253.

I've been through these proofs and understand them however I want to know how many people must be in a room before the chances of at least two people sharing a specific birthday given at the start of the problem (e.g. 'your' birthday) is greater than half.

Clearly the probability of this $=1-P($one or fewer people have this specific birthdate$)=1-P($one person shares this birthdate$)-p($no one shares this birthdate$)$.

I've gone through the methods used to calculate the original problems but every answer I get becomes unsolvable when made greater than 0.5, how would I do this?

• If the date is fixed, the number of people having birthday at this date (let us call it $X$) is binomial-distributed. Just calculate $P(X\ge 2)$ and check for which $n$ it exceeds $\frac{1}{2}$ – Peter Jan 26 '18 at 14:02
• As the answer shows, the result is $613$ – Peter Jan 26 '18 at 14:09

Assuming a 365-day year, the probability you wrote is computed for a given number of people $n$ as $$1 - n\left(\frac1{365}\right)\left(\frac{364}{365}\right)^{n-1} - \left(\frac{364}{365}\right)^{n}$$
Plugging that into WolframAlpha gives that a minimum of $613$ people are required.