Suppose I have a fleet of two-car trains riding around, and that each car is equipped with a data recording device. Unfortunately, some of the recording devices aren't working. I don't know either the exact size of the fleet or the percentage of failed recording devices. I'd like to make a reasonable guess about how many cars I'm missing data from.
- Total fleet size $S$ is unknown.
- Failure rate (failed units / total cars) $F$ is unknown.
- I have data from both cars from $S_2$ trains and from exactly one car from $S_1$ trains.
- Therefore, I know that there are at least $S_1$ failed units. What I don't know is $S_0$, the number of trains with failed units on both cars.
- Let's assume that the distribution of cars with failed units is random, and whether one car has a failed unit is independent of whether it's mate does.
Does the following make sense for a first-order approximation?
- Guess that the failure rate $F'$ is equal to the proportion of missing cars that I know about to total cars that I know about: $F' = S_1 / (2*(S_2 + S_1))$
- Assume that the likelihood of a train having two failed units is $F'^2$.
- Therefore $S_0 = F'^2 * (S_0 + S_1 + S_2)$
- Therefore $S_0 = (S_2 + S_1) * F'^2 / (1 - F'^2)$