It's perhaps worth pointing out here that the BBC currently produces rather little of its content on its own. This wasn't always true, but the costs of producing television have gone up far more than the usage fees that citizens pay in the UK to support the BBC. Thus, most productions are at least co-productions with some other company (often several companies). I imagine that this has significant implications for whatever share of profits the BBC itself enjoys.
UPDATE: I decided that, now that we're specifically talking about profits, the rules have rather changed from what I was originally talking about (not assigning blame here. If the goalposts have moved, that's at least as much from my own assumptions as down to anything anyone said, so I'm happy to accept my share of blame here). To that end, I've looked up
Death in Paradise, and can now confirm that the BBC is
not even one of the production companies behind it (there are a few... just none are the BBC). The BBC does commission the show, much as our networks in the US will commission shows that they don't produce. They do
air the show, so there may well be income generated from shows like
Death in Paradise, but I'm hesitant to go on any limb to suggest how that would work if they're not producing the show, and the network doesn't sell commercial advertising space.
For shows in which the BBC does have definite ownership (at least in part), I refer to this page
https://en.wikipedia.org/wiki/BBC_Studios_Productions
Shakespeare & Hathaway: Private Investigators is quite popular, and the
Strictly Come Dancing franchise (which gives us
Dancing with the Stars) is an undeniable powerhouse. But there will be admittedly few shows that folks who are aren't specifically fans of British television will be aware of. If you watch PBS a lot, you'll definitely recognize some names. But that's how it has almost always been for British television, even Doctor Who. Most of it is on PBS when folks in the US are watching it.
