The problem is as follows:
There are 10 shooters at a shooting range. Each shooter is given 5 bullets. They all begin shooting at 9am and end shooting at 10am, They each shoot all 5 of their bullets over that period.
The shooters each choose 5 random time points over the hour and they shoot 1 bullet per time point. The time resolution used is 1 second (3600 possible time points). So the times 09:37:53.100 and 09:37:53.707 are considered to have occurred in the same second.
What is the probability that two shooters fire a bullet at the same time point?
I thought the birthday problem could be used to solve this problem. I replaced the number of days in a year (365) with 3600, then I replaced the number of people in the room with the total number of bullets fired which is 50, resulting in a roughly 29% probability of two shots occurring at the same time.
However when I reduce the problem to just one shooter with 5 bullets, the above construction gives 0.13% probability.
This is wrong because a shooter can only shoot 1 bullet at a time, the probability should be zero, hence the above method is wrong.
I was wondering if some could suggest a way to solve this particular problem.
I think a rewording of the problem could be as follows:
There is a room with MxN people, where M < N. Each person is associated with only one of M clubs and all the people are equally divided amongst the M clubs. What is the probability that two people from different clubs share the same birthday? (day/month combination only)
How would one apply Poisson distribution to this problem? As it seems that the math gets pretty complicated for the exactly 2 collisions problem.