Very nice on the expected losses on a single night. I would be interested in knowing the expected losses over the entire season as well. Great post!
As postulated, the expected losses per season could be considered as 27x1.2 and thus 33. Erroring on the side of comservativsm but not doomsday prophesy, average losses for a year = 99 crops to crows. (I apologize in advance for the wall of text to follow. I like to be thurough in case folks want to check my math.)
Now, the math used in posts above assumed 0 scarecrow coverage. 30% success and 70% failure as give in the wiki. All fields are planted the whole time, which means eaten plants get replanted with another crop - even if just to preserve the tilling for the next season.
To alter the formula to account for some scarecrow coverage and some not the formula will change as such.
chance of bird eating = E= 30% x ( 1- (Safe Crops / Total Crops )) -assuming no empty tilled fields because we are being efficient. 10 attemtps doesn't matter as the first will either eat or flee.
chance of bird not eating is N = 100% - chance of bird eating.
To simplify the field count (and reduce the magnitude of digits) I am going to measure the fields in units of full scarecrows. S is now Safe Crops = Scarecrows and T = Total Crops and also Total Scarecrows required. Giving us E= 30% x (1 - S/T) and T = 1-E
Most folks on the other forums who post figure on 20 quality sprinklers inside one scarecrow radius. That is 160 fields per scarecrow.
Scarecrows cover 248 fields, but since we are talking many, many scarecrows in each scenario, I'd hesitate to use this number due to watering constraints.
I see quality sprinklers cover 88.88% of the ground they occupy so 220 fields as a scarecrow is possible.
Iridium sprinklers cover 96% of the ground they occupy so 238 is an option.
Regular sprinklers are 80% coverage or 198.
My current preferred layout is 181 fields planted in each scarecrow. 20 quality with 8 spaces under coverage and used, 3 quaility with 7 spaces covered and used.
Use whatever number you like, just for these calculations we must assume all scarecrows must cover equal amounts of fields - or you have to use actual field counts and the numbers are larger.
So where to go with this?
The formula for calculating expected crop loss due to 4 crows is as follows.
chance of bird eating = [E] chance of bird not eating [N]
For a single night the % chance of 4 losses is (E^4 ) x 1 combination= A%
For a single night the % chance of 3 losses is ((E^3)*N) x 4 combinations = b% x 4 = B%
For a single night the % chance of 2 losses is ((E^2)*(N^2))x 6 combinations = c% x 6 = C%
For a single night the % chance of 1 loss is (E*(N^3)) x 4 combinations = d% x 4 = D%
For a single night the % chance of 0 losses is (N^4) x 1 combination = Z%.
where A+B+C+D+Z sums to 100% .
the average of those are the Expected Losses per day. Xd. and 27days*Xd = Xs equals seasonal losses.
Each scarecrow comes with the "cost" of 3 harvests lost per scarecrow as already given up or conceeded. Gs.
The goal is to minimize Gs+Xs.
I have a spreadsheet if anyone wants to see the tons of scenerios I checked or to check your own, but let me just hit the highlights here.
Xs without scarecrows is 33, Gs without scarecrows is 0. So to make scarecrows valuable is for Total losses to be less than 33 or the scarecrow is not worth it. With me so far?
The next step to evaulate if Gs+Xs can ever be less than 33 we'll need to pick some field counts and ratios of coverage.
Let's start small with 181 crops and call than 1 scarecrow's worth of fields. T=1
If all are contained inside the radius S = 1 and T = 1.
E = 30% x ( 1 - 1/1) = 0% N = 1-N = 100%
A% = 100%
B% = C% = D% = Z% = 0%.
Thus Xd = 0.
Xs = 0.
Gs = 3xS = 3x1 = 3.
Xs+Gs = 0 + 3 = 3 for 181 fields fully covered by a scarecrow. That is 30 crops better than 33 our scarecrow-less benchmark. I suggest if you can fit all crops under 1 scarecrow you are better off having the scarecrow than not. By 30 harvests per season.
Let's move up to 362 crops and call than 2 scarecrow's worth of fields. T=2
If all are contained inside the radius S = 2 and T = 2.
E = 30% x ( 1 - 2/2) = 0% N = 1-N = 100%
A% = 100%
B% = C% = D% = Z% = 0%.
Thus Xd = 0.
Xs = 0.
Gs = 3xS = 3x2 = 6.
Xs+Gs = 0 + 6 = 6 for 362 fields fully covered by a scarecrow. That is 27 crops better than 33 our scarecrow-less benchmark. I suggest if you can fit all crops under 2 scarecrow you are better off having the scarecrow than not. By 27 harvests per season.
If you would prefer not to obtain that second scarecrow but instead just use the single you started covering 181 and leave the other half (181) uncovered: S = 1 and T = 2.
E = 30% x ( 1 - 1/2) = 15% N = 1-N = 85%
A% = 52.20%
B% = 36.85%
C% = 9.75%
D% = 1.15%
Z% = 0.05%.
Thus Xd = 0.6
Xs = 0.6x27 = 17
Gs = 3xS = 3x1 = 3.
Xs+Gs =17 + 3 = 20 for 362 fields half covered by a scarecrow. That is 13 crops better than 33 our scarecrow-less benchmark but 14 worse than two scarecrows. I suggest if you can fit all crops under 2 scarecrow you are better off having both the scarecrows, but even 1 scarecrow covering half is better than none.
3 full of Three (543 fields) is better to field 100% covered.
as is 4 of 4, 5 of 5, all the way up to 10 scarecrows or 1881 fields.(In my current method of building them)
Gs at 11/11 is 33, Xs at 11/11 is 0. = 33
Gs at 0/11 is 0, Xs at 0/11 is 33. same / same. But you already have 10 scarecrows maybe, from before. Or 6. Or any number.
at T = 11 scarecrows required, all combinations of Gs + Xs = 33. So you can launch to 11 from anywhere previous and beat the benchmark.
Once you reach the total (or configuration) of fields that requires 12 scarecrows in where Gs+Xs becomes 36 in all instances of S less than 14. And as such one might prefer to remove all scarcrows from the farm.
At T=13 . All instances of adding a scarecrow to your count increases Gs faster than it decreases Xs, so the losses just increase the more scarecrows you place.This holds true for all T greater than 13 as well.
But, this was assuming all scarecrows must be built on tillable tiles. Is this true? Could one build scarecrows on non tillable tiles? I've not looked into that yet. I know that would defeat the abilty to count all fields in terms of equal scarecrow units, but at that point any field covered by the non harvest producing scarecrow would serve to increase S/T which would then then reduce E and thus the Xd and Xs while not increasing Gs. Thus scarecrows have value again.
I'll stop here for a breather then come back for deluxe Scarecrow and farm sizes.
{thanks for getting this far]