AA Fire Tables

[rkr1958]

So I've been down a fun rabbit hole the past two days generating AA fire tables. Below is the table from MWIF and was/is my guide to verification of the tables that I generated using Microsoft EXCEL. Specifically, I wanted the pdf (probability distribution functions) for the 26 AA columns (i.e., -1/6,-1/5, ..., 5/5) for some analysis that I was doing and will do in my current game and AAR. In all the games I've played I've never seen AA fire on a column more than +1/2 or maybe +1/3 but generated did generate the pdfs for all 26 columns.

Forgive me if I'm explaining the obvious but the pdf for each column gives the probability versus the AA fire result (i.e., 1,2,3, ...). Also, not to state another obvious but the convention -X/Y means make Y D10 rolls and sum the lowest X and X/Y means make Y D10 rolls and sum the largest X of them for your AA fire result.

[rkr1958]

So first let me explain how I generated my tables. Maybe I'm missing something but I didn't see a straightforward way to calculate all the pdfs using probability theory, which is exact, except for the easy cases (e.g., -1/X, 1/1, 2/2). So I resorted a Monte Carlo approach where I generated 6 columns of 10,000 D10 "random" samples, which is 60,000 D10 random rolls. In the screen shot of my EXCEL tab below those samples are in columns A-F, rows 2-10001.

I then used those sample to generate my Monte Carlo samples for each of the 26 AA columns. The minimum number of samples for any AA column was 10,000 (e.g., -1/6,-1/5,-1/4). However; where possible I would used the data to "generate" more samples. The most was for 1/1 in which I got 60,000 samples. For AA columns such as -1/3 or +1/3 I could get 20,000 samples. Hopefully, if interested and care, you're following me here.

[rkr1958]

Initially, I generated my 60,000 D10 random rolls using EXCEL's random data analysis discrete random variable generation function. While this worked ok I wasn't happy because the sample pdf of the D10 produced vs the actual was off more than I liked. Also, it took a bit of time even with my new PC to generate these 60,000 discrete samples. But that was only once and if I had been satisfied with the sample the time to generate them would have been ok. But I wasn't. Specifically, it was "expected" that 6000 samples out of the 60,000 total would be generated for each of the 10 die rolls. I accept that in reality you aren't going to match the expected very often. However, I don't like being off by almost 100 samples, or 0.1% as was the case for D10=2. Also, for the astute the sample pdf should sum to 1 across all rolls but didn't. I found that out of the 60,000 samples generate there was 1 zero. In previous 5 sets I generated I found 5 0's in one of those sets. Even after I manually corrected this by replacing the 0(s) was an actual D10 roll(s) I still wasn't happy with the noise (i.e., D10 sample pdf versus actual pdf). So I chosen another approach for generating my random samples. One that would guarantee 6000 samples for each of the D10 rolls.

[rkr1958]

Well maybe not exactly Kosher I feel it was good enough and did guarantee me 6000 samples for each D10 roll in the 60,000 generated. Column A contains a "deck" of 6000 x 1-10 D10 die rolls. That's 10,000 rows. In column B I used EXCEL's uniform random number generator to generate 10,000 pseudo-random number between [0,1). I then copied and pasted as text columns A & B into columns C & D and sorted columns C& D together by the values in column D from smallest to largest. I then copied C and pasted it to column E. I did that 5 more times each time pasting column C to F,G,H,I and J to generate my 6 x 10,000 (or 60,000) D10 random samples, which I then copied and pasted into columns A-F, rows 2-10,001 of my raw data.

[rkr1958]

AA Fire PDF Tables. (1 or 2 of Y).

[rkr1958]

AA Fire PDF Tables. (3,4 or 5 of Y).

[rkr1958]

1. EXCEL spreadsheet attached.

2. The avg for each of the 26 AA columns that I generated were within the 90% confidence interval of the MWIF Avg values in the table of the first post of this thread.

3. The 90%+/- row is that value. The AvgLCI & AvgUCI is the lower and upper 90% confidence interval of the Avg generated from my samples.

4. I think both the average (Avg) and standard deviation (sig) matched that of the MWIF table which gives me some confidence in the pdf distributions generated for each PDF column.

5. The mode, which is the most frequent value, did differ some from the MWIF but the probabilities in those cases were very close, <.001, between MWIF's and mine when they did.

6. The SumErr row is the sum of the pdf - 1, which should be 0. That is, a pdf function should sum to 1 or it isn't a correct pdf function! So, the SumErr row was another check for me.

7. As always, and often, my work like this is subject to error so please let me know if you find any.

8. Also, as always, positive and negative feedback is welcomed.

[paulderynck]

There are places in the RAW7 AA table where higher AA is less effective than the next lowest AA. These were corrected in CE.