Can high-defence settlements in Fallout 4 be attacked?

I've read that high-defence settlements (where the defence rating is higher than the combined food & water value) will not be attacked. Elsewhere, I have read that even high defence settlements can occasionally be attacked. Which is true? What is the maths/algorithm behind the chance of a settlement being attacked?

I think it would be a shame to create a very high defence settlement, armed to the teeth, if it were never going to be attacked so that you could witness those defences in action.

Yes, they can be attacked.

There's a reddit thread where this is discussed, including a formula. The thread is 4 months old, so it's possible this has been tweaked in a patch since then.

I've copied over the formula for raid chance and associated notes, with some formatting improvements:

How Settlement raids work:

1. There are apparently some quests and triggers (I don't have a list of all these triggers) that can force an attack, which will ignore everything I'm about to tell you.

2. A Settlement will only be attacked if it has at least one population.

3. A Settlement will not be attacked more than once every seven days.

4. Every day, there's a chance for your Settlement to be attacked:

Base chance: 2%

Resources add to that: (food + water) / 10 %
(Note that food and water items in your workshop inventory count towards these totals. Thanks to /u/matzman666 for pointing that out.)

Defense and population subtract from that: defense + (population / 2) %

Chance of attack never goes below 2%

Settlement with 50 food + water, 10 defense, and 10 settlers = 2 + 5 - 10 - 5 = -8 => 2% (min value)

Settlement with 200 food + water, 0 defense, and 2 settlers = 2 + 20 - 0 - 1 = 21%

So the chance of attack is always at least 2% per day (unless it's already been attacked in the last week or there are no settlers), but it's not hard to get it that low.

My settlements with a defense rating higher than or equal to the combined resource production have definitely been attacked before. However I'm not sure of the algorithm to calculate how much more likely it is that they will be attacked.

One thing that will almost ensure attacks on settlements, if you have the Wasteland Workshop DLC, is having occupied animal cages at a settlement.

I have an arena that contains around 15 cages filled with animals, and that settlement gets attacked most times that I visit, and is constantly being attacked when I am away from it, despite having a defense rating of over 400 and a resource production a little over 100.

Only the type of enemy trapped inside a cage will attack your settlement, so deathclaws will attack a settlement with a filled deathclaw cage and gunners will attack somewhere that has trapped a gunner.