I don't know if anyone has done this in the past, but all this talk about OBA has gotten me wondering. What are the probabilities for battery access? Actually, the math is simple.[Ed: Ok, I feel pretty stupid now. ;-)]
If you have B black chits, and R red chits, you can arrange them in (B+R)! / (B! R!) different sequences. In order to get N fire missions out of the chits, you need a sequence of N black chits and one red chit, one red chit, then the remaining chits in any order. There are (N+1) (B+R-N-2)! / ( (B-N)! (R-2)! ) different ways to do this. So the chance of getting exactly N fire missions is: (N+1) (B+R-N-2)! B! R! / ( (B+R)! (B-N)! (R-2)! ).
N Fire Missions:
| Black | Red | Expected | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 11 | 3 | 5.50 | 3% | 6% | 8% | 10% | 11% | 12% | 12% | 11% | 10% | 8% | 6% | 3% |
| 10 | 2 | 6.67 | 2% | 3% | 5% | 6% | 8% | 9% | 11% | 12% | 14% | 15% | 17% | |
| 10 | 3 | 5.00 | 2% | 7% | 9% | 11% | 12% | 13% | 12% | 11% | 9% | 7% | 7% | |
| 9 | 2 | 6.00 | 2% | 4% | 5% | 7% | 9% | 11% | 13% | 15% | 16% | 18% | ||
| 9 | 3 | 4.50 | 5% | 8% | 11% | 13% | 14% | 14% | 13% | 11% | 8% | 5% | ||
| 9 | 4 | 3.60 | 8% | 13% | 15% | 16% | 15% | 13% | 10% | 7% | 4% | 1% | ||
| 8 | 2 | 5.33 | 2% | 4% | 7% | 9% | 11% | 13% | 16% | 18% | 20% | |||
| 8 | 3 | 4.00 | 5% | 10% | 13% | 15% | 15% | 15% | 13% | 10% | 5% | |||
| 8 | 4 | 3.20 | 9% | 15% | 17% | 17% | 15% | 12% | 8% | 5% | 2% | |||
| 7 | 2 | 4.67 | 3% | 6% | 8% | 11% | 14% | 17% | 19% | 22% | ||||
| 7 | 3 | 3.50 | 7% | 12% | 15% | 17% | 17% | 15% | 12% | 7% | ||||
| 7 | 4 | 2.80 | 11% | 17% | 19% | 18% | 15% | 11% | 6% | 2% | ||||
| 6 | 2 | 4.00 | 4% | 7% | 11% | 14% | 18% | 21% | 25% | |||||
| 6 | 3 | 3.00 | 8% | 14% | 18% | 19% | 18% | 14% | 8% | |||||
| 6 | 4 | 2.40 | 13% | 20% | 21% | 19% | 14% | 9% | 3% | |||||
| 5 | 2 | 3.33 | 5% | 10% | 14% | 19% | 24% | 29% | ||||||
| 5 | 3 | 2.50 | 11% | 18% | 21% | 21% | 18% | 11% | ||||||
| 5 | 4 | 2.00 | 17% | 24% | 24% | 19% | 12% | 5% |
From this table, it is easy to see how many fire missions each of the different nationalities will average.
Allied Minor 3
American 5
Axis Minor 3
British 5.33
Chinese 3.33
or 4 or 2.5
Finnish 3
French 4
German 4
Italian 3.5
Japanese
3.33
Russian 3.33
The best you can get is British OBA with plentiful ammo and pre-registered hex(es), which will average 6.67 fire missions. On the other hand, Chinese 3-3-7's with scarce ammo will only average 2.00.
Once you draw the first red chit, you may want to know how many more fire missions you can expect. Or, you may want to know how many fire missions to expect before they get interrupted by the first red chit. Again, you have B black chits and R red chits. To get N fire missions, you need a sequence of N black chits, a red chit, then the remaining chits. There are (B+R-N-1)! / ( (B-N)! (R-1)! ) ways to do this.
N Fire Missions Until a Red Chit:
| Black | Red | Expected | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 11 | 3 | 2.75 | 21% | 18% | 15% | 12% | 10% | 8% | 6% | 4% | 3% | 2% | 1% | 0% |
| 11 | 2 | 3.67 | 15% | 14% | 13% | 12% | 10% | 9% | 8% | 6% | 5% | 4% | 3% | 1% |
| 10 | 3 | 2.50 | 23% | 19% | 16% | 13% | 10% | 7% | 5% | 3% | 2% | 1% | 0% | |
| 10 | 2 | 3.33 | 17% | 15% | 14% | 12% | 11% | 9% | 8% | 6% | 5% | 3% | 2% | |
| 10 | 1 | 5.00 | 9% | 9% | 9% | 9% | 9% | 9% | 9% | 9% | 9% | 9% | 9% | |
| 9 | 4 | 1.80 | 31% | 23% | 17% | 12% | 8% | 5% | 3% | 1% | 1% | 0% | ||
| 9 | 3 | 2.25 | 25% | 20% | 16% | 13% | 10% | 7% | 5% | 3% | 1% | 0% | ||
| 9 | 2 | 3.00 | 18% | 16% | 15% | 13% | 11% | 9% | 7% | 5% | 4% | 2% | ||
| 9 | 1 | 4.50 | 10% | 10% | 10% | 10% | 10% | 10% | 10% | 10% | 10% | 10% | ||
| 8 | 4 | 1.60 | 33% | 24% | 17% | 11% | 7% | 4% | 2% | 1% | 0% | |||
| 8 | 3 | 2.00 | 27% | 22% | 17% | 13% | 9% | 6% | 4% | 2% | 1% | |||
| 8 | 2 | 2.67 | 20% | 18% | 16% | 13% | 11% | 9% | 7% | 4% | 2% | |||
| 8 | 1 | 4.00 | 11% | 11% | 11% | 11% | 11% | 11% | 11% | 11% | 11% | |||
| 7 | 4 | 1.40 | 36% | 25% | 17% | 11% | 6% | 3% | 1% | 0% | ||||
| 7 | 3 | 1.75 | 30% | 23% | 18% | 13% | 8% | 5% | 3% | 1% | ||||
| 7 | 2 | 2.33 | 22% | 19% | 17% | 14% | 11% | 8% | 6% | 3% | ||||
| 7 | 1 | 3.50 | 13% | 13% | 13% | 13% | 13% | 13% | 13% | 13% | ||||
| 6 | 4 | 1.20 | 40% | 27% | 17% | 17% | 10% | 5% | 2% | 0% | ||||
| 6 | 3 | 1.50 | 33% | 25% | 18% | 12% | 7% | 4% | 1% | |||||
| 6 | 2 | 2.00 | 25% | 21% | 18% | 14% | 11% | 7% | 4% | |||||
| 6 | 1 | 3.00 | 14% | 14% | 14% | 14% | 14% | 14% | 14% | |||||
| 5 | 4 | 1.00 | 44% | 28% | 16% | 8% | 3% | 1% | ||||||
| 5 | 3 | 1.25 | 38% | 27% | 18% | 11% | 5% | 2% | ||||||
| 5 | 2 | 1.67 | 29% | 24% | 19% | 14% | 10% | 5% | ||||||
| 5 | 1 | 2.50 | 17% | 17% | 17% | 17% | 17% | 17% | ||||||
| 4 | 3 | 1.00 | 43% | 29% | 17% | 9% | 3% | |||||||
| 4 | 2 | 1.33 | 33% | 27% | 20% | 13% | 7% | |||||||
| 4 | 1 | 2.00 | 20% | 20% | 20% | 20% | 20% | |||||||
| 3 | 3 | 0.75 | 50% | 30% | 15% | 5% | ||||||||
| 3 | 2 | 1.00 | 40% | 30% | 20% | 10% | ||||||||
| 3 | 1 | 1.50 | 25% | 25% | 25% | 25% | ||||||||
| 2 | 3 | 0.50 | 60% | 30% | 10% | |||||||||
| 2 | 2 | 0.67 | 50% | 33% | 17% | |||||||||
| 2 | 1 | 1.00 | 33% | 33% | 33% | |||||||||
| 1 | 3 | 0.25 | 75% | 25% | ||||||||||
| 1 | 2 | 0.33 | 67% | 33% | ||||||||||
| 1 | 1 | 0.50 | 50% | 50% |
The "0%" entries in this table are really probabilities < 0.5%.
Of course, now that you know the formulas, you can keep track, chit by chit, of your odds. And garner some very strange looks from your opponent. And never forget the intimidation/whining potentials from such a table :) .
Scott