I have survey data of interviews with people at certain outdoor recreation sites. The survey records the amount of time each interviewer spends at a site, and each person surveyed at the given site is asked how long they plan to spend at the site. So, I know the amount of time in minutes we interview people at certain sites, and (roughly) how long each person interviewed stayed at the site. I am assuming that there are 12 hours of daylight that people can use to go a site.
I'm trying to figure out the probability that an interviewer and any given interviewee were at the same site at the same time. I' not sure how to obtain a general formula for this as a function of the data I have, but here's how I've been thinking about it:
As a simplified example, let's once again assume that there are 12 hours of possible time that people can visit a site. Say that I interviewed people for 4 hours. One person interviewed told me that he or she is spending 7 hours at the site that day. Using that information, I want to know the probability that we were at the site at the same time.
I'm assuming for convenience that people only enter and leave the site on the hour. There are 9 different ways that I could have visited the site with my 4 hours (hours 1-4, hours 2-5, hours 3-6, hours 4-7, hours 5-8, hours 6-9, hours 7-10, hours 8-11, hours 9-12). There are 6 different ways the interviewee could have visited with his or her 7 hours (hours 1-7, hours 2-8, hours 3-9, hours 4-10, hours 5-11, hours 6-12).
9 times 6 is equal to 54 possible time arrangements. From back-of-the envelope calculations, I've calculated that 48 of these possible time arrangements include an overlap in when the interviewer and interviewee are at the site. So the probability that they're there at the same time is 48/54 = 8/9.
Is there any way to generalize this in terms of a formula? I have a lot of observations, and I can't possibly do these calculations for each one. I've been looking for patterns in the examples I'm doing myself, but every time I think I have a solid formula, it doesn't work on the next example I try.