"Fun" With statistics and noble probability

[DeletedUser107795]

Woohoo! Statistics! Best part of TW!


I thought I would try out some math concepts to determine noble probability. I rooted around in the forums and found some half-assed math. Wouldn't surprise me if you could find this somewhere else. (Let me have my fun you trolls! I have feelings!!)

Anywho,

Using the fun-inspired central limit theorem. We can say that any random variable with n =+ 30 is normally distributed.
This means we can narrow down the average to a more realistic number.

Here's the results:

Probability of a single noble hitting 25 or above = 70 percent

Using the empirical rule, we know that 68% of the sample falls between 1 standard deviation of the mean. (95 within two, 100 within 3)

there is an appox. 68% chance your noble will fall between (22.84 - 32.42)
there is an approx. 27% chance your noble will fall between (18.24 - 22.84, 32.42 - 37.02)**

**This is slightly nonsensical for the game as it goes past noble boundaries. However you can assume there is
roughly a 13% that you will be below 22.84 or above 32.42. (13.5+13.5=27)


So this is close, but not exact. Anyone got better?




Want to check my math? Here we go!


Pulling up Excel, I generated random numbers from 20-35. 30 rows X 100 columns to be safe. (Thats 3000 random numbers between 20-35)

Average(U): 27.44
STDEV: 4.6

So, Now we need our Z formula. (X-U/STDEV)

p(X>25)
=(25-27.44)/4.6
=-0.53

Pulling up our table of standard values, -0.53 = 0.2981
1-0.2981 = 0.7019

Therefore the chances your noble being above 25% is 70%

However, This isn't quite right as you need 4 nobles to equal 100+.

So, Using the empirical rule, we know that 68% of the sample falls between 1 standard deviation of the mean.

That means
27.44 - 4.6 = 22.84
27.44 + 4.6 = 32.042

So there is a 68% chance your noble will fall between (22.84 - 32.42)
there is a 27% chance your noble will fall between (20 - 22.84, 32.42 - 35)

 

[DeletedUser]

Now aggregate the numbers to show the probability of a noble train reducing loyalty by 100+ (add sensitivity analysis in case you want to split up nobles to pair them with separate nukes that hit at different times to confuse the opponent, thereby giving percentages of success for different gaps of time between noble 1 and noble 4 hitting).

For instance, a noble train with a 4 hour gap between 1st and last noble has x% chance of being successful, while a 2 hour gap has x%, etc. Of course, these occurrences are random (previously stated) and independent (wasn't stated, but your work seems to have incorporated this).

And yes, the math is the best part of the game.

 

[DeletedUser]

why would loyalty removed be normally distributed? wouldn't it just be 1/16th chance of 20 removed, 1/16th chance of 21 removed... etc

 

[DeletedUser]

It's not distributed as a normal distribution as far as I know. I'm pretty sure it's uniform distribution.

Here are the exact numbers.

Sorry for bumping in. I was looking for the calculations for those numbers as we've got a fancy smancy world starting on the Norwegian servers now with loyalty in the range [20, 50]. Yeah, fun times. So I was planning to calculate how probable it is that a village would get taken with X number of nobles. But that would be a pain to calculate. So I made a simple random generator in Python to do it for me. I simulated 10 000 000 noble trains. If anyone is curious, the probability for that loyalty intervall was approximately: Two nobles: 0.1 % - Three nobles: 63.5 % - Four nobles: 35.9 % - Five nobles: 0.96 %. If anyone's curious about that, send me a message.