what is the probability of other person having the same birthday as yours in a group of 70 people? given I know my birthday?

Question

I understand the birthday problem wherein the probability of 2 people having the same birthday in a room of 70 people is 99.9% but what about the probability of any person in the same room having my birthday? is it the same or is this a conditional probability problem?

Answer 1

It is not the same because the event "a person has mybirthday" is strictly included in the event "two people have the same birthday".

You want to know if, among $70$ people, one has your birthday. Let's compute the probability of the complementary, that is the probability that none of the $70$ people has your birthday. For one person, the probability is $(364/365)$. So for $70$, you get $\left( \frac{364}{365} \right)^{70}$. You deduce that the probability that one people has your birthday is $$P = 1 - \left( \frac{364}{365} \right)^{70} \simeq 17 \% $$

(I did not take into account the fact that some years are bissextiles...)

Answer 2

In the tradition birthday problem each person can not share a birthday with anyone else. With each new person a potential birthday is eliminated decreases the probability for the next person. Example the probability of the $2$nd person not having one of the previous birthdays is $\frac {364}{365}$ but the probability of the $20$th person (assuming somehow the first $19$ people are all different) is $\frac {346}{365}$ and of the $n$th person is $\frac {365-(n-1)}{365}$.

That is a lot different than not sharing a single birthday. Each new person simply doesn't have to have your birthday and that never changes. So the probability for each new person not having your birthday is $\frac {364}{365}$ and it doesn't matter if it is the $2$nd person or the $70$th person.

In the traditional problem the probability of everybody have different birthdays is $\frac {364}{365}*\frac {363}{365}*.....*\frac{296}{365}=\frac {364!}{295!*(365)^{69}} \approx 8.4\times 10^{-4} ... really small...$

To have $69$ people not have your birthday is $\frac{364}{365}*\frac{364}{365}*\frac{364}{365} * ...*\frac {364}{365} = (\frac {364}{365})^{69} \approx .83$. That's a huge difference.