We're about to site bees on an allotment (not ours, sadly). Harrow Council have been very helpful.
So, say my plot neighbour put a bee hive on her plot. What's the chance of me getting stung just walking next by it.
The local BKA will advise on siting. If she were to put the hive so that the entrance was adjacent to and facing the communal path, then walking directly in front of the hive will get you stung sooner rather than later. Especially if you wear perfume or cologne.
But if the entrance faces away from paths, and there are (say) fruit bushes in front of it to force the bees to fly over them, then you're at no more risk than you are from wasps' nests nearby. And there are bound to be dozens of them.
What about when I strim the paths.
Shouldn't be e problem. I've used strimmers and lawnmowers on the local BKA apiary around 30 or so hives, strimming right up to the hive stands. The bees are quite unfussed. Obviously, if you knock into the hive, you'll alarm them.
Assuming my plot neighbour keeps the bees fairly inexpertly, what's the chance of the bees swarming,
100%, assuming varroa or some other problem doesn't kill them first.
and if they swarm what's the chance of them stinging lots of people badly.
Zero. Bees are at their calmest when they swarm. They're full of honey for the journey, and programmed to set up their new home. I guess that if you see a swarm, typically hanging on a tree nearby where they regroup prior to flying to their new location, and you beat at it with a stick, then you'll get a response.
And if the bees are kept inexpertly, what's the chance of the colony actually adding to the veroa problem?
Sadly, varroa is endemic. so it won't add to the problem.