Actually, what I did to get the chance of a bomber shot down was to run 6 bombers against an AA gun on Frood. (for one round of combat)
34.36%6 Bomno units.0 IPCs
40.02%5 Bom1 Bom15 IPCs
19.59%4 Bom2 Bom30 IPCs
5.1%*******3 Bom3 Bom*45 IPCs
0.83%2 Bom4 Bom60 IPCs
0.1%1 Bom*****5 Bom75 IPCs
Now, technically, just because you have a 1 in 6 chance to shoot down a bomber does not mean you will always get a positive result for every six bombers attacking. The chances are actually 6 take 1 or 100%- 6 take 5 that you will score a hit. On any given roll you have a 16.67% chance of hitting a bomber, but it does not scale like that in statistics.
Therefore, letting Frood run the numbers for me, so I don’t have to go through the binomial distribution to figure them out for myself!, the odds of losing 1 bomber or no bombers is the odds of losing no bombers + the odds of losing one bomber or:
34.36%+40.02% = 74.38%.
74.38% of 15 IPCs = 11.157 IPCs, but since you cannot lose 0.157 IPCs, we round to the nearest number, which is 11 IPCs.
To get the average damage done by 6 bombers we can determine the mean result and multiply that by the number of rolls. The mean (or the expected value of Y) is the sum of all possible results divided by the number of possible results. In this case, 1+2+3+4+5+6= total, total/6 = mean of Y. Therefore the mean of Y is 3.5 IPC per bomber. Since this is a mean and not the actual damage done we can have a partial IPC.
So we know that 74.38% of the time we have 5 or 6 bombers surviving their SBR runs. Therefore, we can calculate the damage done by doing the following:
Number of Bombers (5) * Mean (3.5) = Total Damage (17.5 IPC)
+
Number of Bombers (6) * Mean (3.5) = Total Damage (21 IPC)
Total Damage/2 (for two results) = 38.5/2 = 19.25 IPC in damage.
Now we clear the partial IPC since you cannot surrender only a part of an IPC in this game.
The total damage we can expect to do, in one round, with 6 bombers attacking, 74.38% of the time, is 19 IPC and the total damage we can expect to take, in one round, with 6 bombers attacking 1 anti-aircraft cannon is 11 IPC.
This is an 8 IPC per round advantage to the alliance performing the strategic bombing runs. In this case, it is much more significant as it is 11 IPC inflicted on the axis powers (roughly 5 or 6 IPC each) vs 19 IPC damage done to Russia each round. Considering Russia is only earning between 18-30 anyway, this seriously weakens their ability to maintain a defensive posture forcing them to fall back sooner.
Of course, that’s the theory, and how I derived the numbers I was using.
In case anyone wonders, I am using result for 5 or 6 bombers because the odds of losing two or more bombers to anti-aircraft fire becomes prohibitively small. However, it would be valid to run the numbers using 4 bombers as a possible result as this would happen 25.52% of the time. I just prefer to be a little more optimistic about my bombers and pessimistic about my anti-aircraft cannons.