I found two flies in my house, marked them, then later found two more flies without marks. I think this means the population of flies in my house is infinite.
You want to consider, based on a distribution of potential fly populations, what the odds are that sampling two of them will fail to capture two specific ones. For example, you can state with certainty that the population is above 3.
Applying that directly, we would estimate that the odds of a fly in your house being marked are less than 150%, which tells us that you should be taking larger samples.
(With only two marked flies, this estimate will always be less informative than the fact that you sampled n flies -- it will tell you that the population is probably greater than 2/3 of n, while the sampling procedure tells you that the population is necessarily at least n+2. But as you mark more flies, the estimate will be more informative.)
Mostly because I thought it was fun, a worked example where you mark 100 different flies and then catch 100 unmarked flies (releasing each fly individually immediately after catching it; maybe you caught the same fly 100 times):
By the rule of three, we estimate the probability that a fly in your house is marked at p < 3/100.
We can also model the probability that a fly in your house is marked as 100/n, where n is the number of flies in your house.
Then 100/n < 3/100 and n > 100*100/3 = 3,333. There are probably more than 3333 flies in your house.
This reminds me of the time I found two flies in my house despite the windows being closed. I vacuumed them up (more fun that way) and the next day there were two identical looking flies in the house. Did they get out of the vacuum cleaner? I vacuumed them up again.
The next day, same story. The flies were now appearing throughout the day. They all looked the same. Where the hell were they coming from? The windows were still closed. I kept vacuuming.
Eventually things were starting to get very irritating. I hunted for an entry point for days without finding anything and the flies just kept on appearing. I was pretty good at vacuuming flies by this point.
By day six I spotted a group of them hanging out near my Kentia palm. Aha! A fly had laid eggs in the soil of the plant. I had no idea that it was a suitable food source for larvae. Needless to say I quickly filled it with gravel and I guess that's the story of how I became a qualified Fly Detective.
Another fun thing you can do with flies, and also bees and wasps, apart from vacuuming them up, is put them on a leash. But first you have to freeze them.
Catch one in a cup or plastic bag and stick it in the freezer for about 10 minutes. When you take it out it will look dead, but it's not (unless you leave it in too long.) Being careful not to rip it's wings off, tie a small string or fishing line to one of it's legs.
In a few minutes it will thaw and start to walk around, and then start to fly. You can now walk it around the park like you were carrying a balloon.
Where would you say it falls on the spectrum of animal abuse in relation to going fishing, fly swatting, and walking the dog? Those are all activities I'm personally ok with.
That's scary. I used to be a hardcore insectaphobe, but I got over it over the last couple years. Now I don't care that much when I see insects in my house. But it is within reason! I wouldn't want bugs reproducing in my house, that sounds like a slippery slope!
It's only the size and age of them that makes one fly distinguishable from another. When they're all being born at the same time there's absolutely nothing that makes them distinct, and that's your clue that eggs have been laid somewhere in the vicinity.