Probabilities of blown-away package ownership, based on frequency and distance Yesterday, because of high wind, a package got blown into our backyard.
We weren't sure whether the package belonged to us or to one of our five neighbors.
It occurred to me that if we knew the frequency with which each neighbor receives packages, we might be able to calculate the probability that the package was blown from each house's doorstep.
But because all of our doorsteps are different distances away from where the package was found, we would have to factor in not only the frequency, but also the distance from each doorstep to the location where the package was found.
I know how to compute the probabilities based only on frequency, but I'm not quite sure how to also factor in the distance.  Any tips?
P.S. In case you're curious, although our household probably receives more packages than any of our neighbors, the package that we found in our backyard belonged to the neighbor furthest away from us!
 A: You are asking what is the probability that a package belongs to a certain neighbor given that it is in your yard? If $Y$ is the event a package is in your yard and $X$ is the random variable representing which neighbor it belongs to, then you are asking what is $P(X=i|Y)$, the probability that $X$ is a particular neighbor  $i$ given that $Y$. By Bayes formula,
$$P(X=i|Y) = \frac{P(Y|X=i) P(X=i)}{\sum_kP(Y|X=k)P(X=k)}$$
where the sum in the denominator is over all neighbors in your event space.
Each $P(X=i)$, the probability that a randomly selected package belongs to a certain neighbor regardless of where it is found, might be modeled as the relative frequency, call it $\pi(i)$, at which that neighbor receives packages compared to total package delivery to your neighborhood.
Each $P(Y|X=i)$, the probability that a package for neighbor $i$ ends up in your yard, might be modeled as a function of distance to your yard. A simple model would be that the probability is inversely proportional to the square of distance, that is, $P(Y|X=i) = \alpha / d_i^2$, for some constant of proportionality $\alpha$. However, this constant will cancel out, leaving the model
$$P(X=i|Y) \approx \frac{\frac{1}{d_i^2} \pi(i)}{\sum_k \frac{1}{d_k^2} \pi(k)} $$
Obviously, you could refine this model in all sorts of ways. For example, you might graph out the geometry of your neighborhood, including your yard, and model the path of the wind-blown package as a two-dimensional random walk. Then, the model becomes what is the probability that a random walk from a neighbors stoop is within your yard at a certain time? And then consider when in the day/week each neighbor typically receives packages with respect to when it was windy. You could devise models for wind speed and direction given weather patterns. You could considering fencing and walls both in permissible package paths and their effect in channeling wind. And so forth, and so on. However, it would probably just be easier to read the label.
