My Birthday Problem

Let us consider that we have invited $n$ people.

So the probability that the first person does not share birthday with me is $\frac{364}{365}$ . The same holds for the second,third fourth person etc.

Following this argument the probability that among the $n$ people I have invited no one shares their birthday with me is $(\frac{364}{365})^{n}$ .

So the probability that at least one person shares birthday with me will be $1-(\frac{364}{365})^{n}$ .

We want this to be equal to half and hence we have $1-(\frac{364}{365})^{n} = 0.5$ .

Solving for $n$ we get $n = 253$ to be the minimum integral value for which this holds true hence out answer is $\boxed{253}$ .

The probability that person 1 does not match my birthdate is 364/365. The probability that person 2 does not match my birthdate remains 364/365 and similarly 364/365 so on for person 3 and 4 etc. Multiplying these probabilities, i.e (364/365)^n at n= person 253 the probability of not matching falls below 50%. Thus the probability of matching is above 50 % if i invite 253 ppl.

The probability that NONE of my friends that I invited has the same birthday as me is equal to $1-(\frac{364}{365})^{x}$

So we need to find out a value of n where $1-(\frac{364}{365})^{n}$ = 0.5

$1-(\frac{364}{365})^{n}$ = 0.5

( $\frac{364}{365})^{n}$ = 0.5

log( $\frac{364}{365})^{n}$ = log 0.5

n (log $\frac{364}{365})$ = log 0.5

n= $\frac{log (364/365)}{log (0.5)}$

n = 252.65

So I have to have at least 253 friends over