I've been reading about the birthday problem which, as I'm sure many of you will know, is a statistical problem which aims at finding out the how many people you would need in a random group to be certain that two of them shared a birthday. I've read the wikipedia article and am happy with the concept and the answers to this problem. What I'm interested in doing is expanding the principle. I've been trying to work out the answer to a similar problem, but where you simply wanted to know the probability that two people were born in the same week and in the same month. I'm not really sure how to go about this, though, so my first question is: is there a general equation I can use to extend the problem to these cases? But, I know there are many articles and stackexchange questions on this, so I wouldn't ask unless I had a specific problem, which is this:
Suppose a person has met 500 people in their lifetime. What is the probability that seven of those 500 share a birthday in the same two month period?
I think the answer to my last question is that it's certain. But could I ask what is the smallest number of people you would need for the probability that – in a group of 500 people – the probability of them sharing a birthday in a two month period is less that 50%? If that makes sense?
Okay, thank you everybody, edited to tidy up:
Question 1: What is the smallest group of randomly selected people required such that the probability that two of them share a birthday within one week of each other is at least 75%?
Question 2: What is the smallest group of randomly selected people required such that the probability that two of them share a birthday within thirty days of each other is at least 75%?
Question 3: What is the smallest group of randomly selected people required such that the probability that seven of them share a birthday within sixty days of each other is at least 75%?
Question 4: In a group of 30 randomly selected people, what is the probability that seven of them will share a birthday with fifty days?
I hope that's a lot clearer. I had no idea how to word these questions until I posted this and am grateful to everyone who's contributed for helping me do so 🙂
Best Answer
As suggested by Henry, questions 1 and 2 are answered by one generalization Wikipedia gives for the birthday problem, for near-misses. Take $m=365$, $k=7$ or $k=30$ respectively, and determine the smallest $n$ where $$p(n,k,m)=1-\frac{(m-nk-1)!}{m^{n-1}(m-n(k+1))!}$$ is greater than $0.75$.
If you wanted to count 7 people sharing the same birthday, there is another generalization that is relevant, for multiple collisions.
However questions 3 and 4 are combining both generalizations, asking for a "multiple near-miss". This seems rather tricky, and I don't know if it's been solved exactly in general.