Shields, was Re: Armor (long)

Below are some stats that take into account losses to threshold rolls. I have
selected a Privateer (18 tons), Escort Cruiser (26 tons), and
Battledreadnought (60 tons). This shows the increasing damage protection and
lower cost (per point of protection) as ships increase in size. (Best
displayed in a
mono-spaced font)

Average damage per Die no screens: 0.666666
  level 1:    0.5
  level 2:    0.333333
  level 3:    0.166666

Average dice per damage: no screens: 1.49
  level 1:    2.00
  level 2:    3.00
  level 3:    6.00

Chance to reduce screens at 1st threshold roll:
  no reduction:            83% (5/6,10/12,15/18) (.833333)
  loose at least 1 level:  17% (1/6, 2/12, 3/18) (.166666)
  loose at least 2 levels: 8-11% (1/12,2/18) (.083333-.111111)
  loose all screens:        6% (1/18)            (.055555)
  *NOT cumulative! 17% contains the 8-11% which contains the 6%

Chance to reduce screens at 2nd threshold roll: no reduction: 81% (.805555)
loose at least 1 level: 19% (.194444)
  loose at least 2 levels: 9-12% (.090277-.123457)
loose all screens: 6% (.058641)

Chance to reduce screens at 3rd threshold roll: no reduction: 77% (.767747)
loose at least 1 level: 23% (.232253)
  loose at least 2 levels: 10-14% (.101376-.138698)
loose all screens: 6% (.062080)

Formulas to simulate loss due to threshold rolls: No screens remain constant
Before First Threshold: Figure maximum avg. gain After 1st Threshold: 2nd
Threshold: 3rd Threshold:
	     L1  L2  L3      L1  L2  L3      L1  L2  L3
  No Change: 83% 83% 83%     81% 81% 81%     77% 77% 77%
  -1:        17%  9%  6%     19% 10%  7%     23% 13%  9%
  -2:             8%  5%          9%  6%         10%  8%
  -3                  6%              6%              6%

Subtract the no screen level # dice from the Before First Threshold total for
a level of protection. Multiply the Difference by the No Change%.
  Level One: Multiply Difference by the -1 %, divide by 2, multiply by
1.49. Add
results of -1, and No Change.
  Level Two: Multiply Difference by the -1 %, divide by 3, multiply by
2.
Multiply Difference by the -2 %, divide by 3, multiply by 1.49. Add
results of
-1,-2, and No Change.
  Level Three: Multiply Difference by the -1 %, divide by 6, multiply by
3.
Multiply Difference by the -2 %, divide by 6, multiply by 2. Multiply
Difference
by -3 %, divide by 6, multiply by 1.49. Add results of -1,-2,-3, and No
Change.

Examples:
--------------------------------

Privateer (18 tons) 9 damage: 4,5

To reach the 1st threshold level:
Screen Level - # dice  - dice saved - damage saved
  none       -  5.96   -            -
  level 1    -  8.00   -  2.04      -  1.36
  level 2    - 12.00   -  6.04      -  4.03
  level 3    - 24.00   - 18.04      - 12.03

To reach the 2nd threshold level (destruction):
Screen Level - # dice  - dice saved - damage saved
  none       -  7.45   -            -
  level 1    -  9.57   -  2.12      -  1.41
  level 2    - 14.47   -  7.02      -  4.68
  level 3    - 27.56   - 20.11      - 13.41

Totals
Screen Level - Dice   - Damage - Dice/ - Damage/ - Cost/ - Cost/
               Saved  - Saved  - Tons  - Tons    - Die   - Damage
  level 1    -  4.16  -  2.77  - 1.39  - 0.92    - 6.01  - 9.03
  level 2    - 13.06  -  8.71  - 2.18  - 1.45    - 3.82  - 5.74
  level 3    - 38.15  - 25.44  - 4.24  - 2.83    - 1.97  - 2.95

------------------------------

Escort Cruiser (26 tons) 13 damage: 4,4,5

To reach the 1st threshold level:
Screen Level - # dice  - dice saved - damage saved
  none       -  5.96   -            -
  level 1    -  8.00   -  2.04      -  1.36
  level 2    - 12.00   -  6.04      -  4.03
  level 3    - 24.00   - 18.04      - 12.03

To reach the 2nd threshold level:
Screen Level - # dice  - dice saved - damage saved
  none       -  5.96   -            -
  level 1    -  7.91   -  1.95      -  1.30
  level 2    - 11.57   -  5.61      -  3.74
  level 3    - 22.04   - 16.08      - 10.72

To reach the 3rd threshold level (destruction):
Screen Level - # dice  - dice saved - damage saved
  none       -  7.45   -            -
  level 1    -  9.94   -  2.49      -  1.66
  level 2    - 14.41   -  6.96      -  4.64
  level 3    - 27.30   - 19.85      - 13.23

Totals
Screen Level - Dice   - Damage - Dice/ - Damage/ - Cost/ - Cost/
               Saved  - Saved  - Tons  - Tons    - Die   - Damage
  level 1    -  6.48  -  4.32  - 2.16  - 1.44    - 3.86  - 5.79
  level 2    - 18.61  - 12.41  - 3.10  - 2.07    - 2.67  - 4.03
  level 3    - 53.97  - 35.98  - 6.00  - 4.00    - 1.39  - 2.08

-------------------------------

Battledreadnought (60 tons) 30 damage: 7,7,8,8

To reach the 1st threshold level:
Screen Level - # dice  - dice saved - damage saved
  none       - 10.43   -            -
  level 1    - 14.00   -  3.57      -  2.38
  level 2    - 21.00   - 10.57      -  7.05
  level 3    - 42.00   - 31.57      - 21.05

To reach the 2nd threshold level:
Screen Level - # dice  - dice saved - damage saved
  none       - 10.43   -            -
  level 1    - 13.85   -  3.42      -  2.28
  level 2    - 20.26   -  9.83      -  6.55
  level 3    - 38.58   - 28.15      - 18.77

To reach the 3rd threshold level:
Screen Level - # dice  - dice saved - damage saved
  none       - 11.92   -            -
  level 1    - 15.80   -  3.88      -  2.59
  level 2    - 23.05   - 11.13      -  7.42
  level 3    - 43.66   - 31.74      - 21.16

To reach the 4th threshold level (destruction):
Screen Level - # dice  - dice saved - damage saved
  none       - 11.92   -            -
  level 1    - 15.58   -  3.66      -  2.44
  level 2    - 22.87   - 10.95      -  7.30
  level 3    - 42.82   - 30.90      - 20.60

Totals
Screen Level - Dice    - Damage - Dice/  - Damage/ - Cost/ - Cost/
               Saved   - Saved  - Tons   - Tons    - Die   - Damage
  level 1    -  14.53  -  9.69  -  4.84  - 3.23    - 1.72  - 2.58
  level 2    -  42.48  - 28.32  -  7.08  - 4.72    - 1.17  - 1.77
  level 3    - 122.36  - 81.58  - 13.60  - 9.07    - 0.61  - 0.92

===============================
Summary:

Level 1
Ship Class   - Dice    - Damage - Dice/  - Damage/ - Cost/ - Cost/
               Saved   - Saved  - Tons   - Tons    - Die   - Damage
  Privter    -   4.16  -  2.77  -  1.39  - 0.92    - 6.01  - 9.03
  E Cruis    -   6.48  -  4.32  -  2.16  - 1.44    - 3.86  - 5.79
  BDrdnt     -  14.53  -  9.69  -  4.84  - 3.23    - 1.72  - 2.58

Level 2
Ship Class   - Dice    - Damage - Dice/  - Damage/ - Cost/ - Cost/
               Saved   - Saved  - Tons   - Tons    - Die   - Damage
  Privter    -  13.06  -  8.71  -  2.18  - 1.45    - 3.82  - 5.74
  E Cruis    -  18.61  - 12.41  -  3.10  - 2.07    - 2.67  - 4.03
  BDrdnt     -  42.48  - 28.32  -  7.08  - 4.72    - 1.17  - 1.77

Level 3
Ship Class   - Dice    - Damage - Dice/  - Damage/ - Cost/ - Cost/
               Saved   - Saved  - Tons   - Tons    - Die   - Damage
  Privter    -  38.15  - 25.44  -  4.24  - 2.83    - 1.97  - 2.95
  E Cruis    -  18.61  - 12.41  -  3.10  - 2.07    - 2.67  - 4.03
  BDrdnt     - 122.36  - 81.58  - 13.60  - 9.07    - 0.61  - 0.92

> At 12:21 PM 11/8/96 EST, Brian Bell wrote:
<<snip>>
> Chance to reduce screens at 1st threshold roll:

I don't think your stats are correct. Take the "no reduction" line above.
With one shield there is a 5/6 chance that the ship will not lose a
shield. However, with 2 shields, the possibility of not losing a shield is not
10/12, it is 25/36. With two shields you roll two dice (one per shield)
meaning that there are 36 possible dice combinations. You lose one shield in
any of the following combinations: 1,6; 2,6; 3,6; 4,6; 5,6; 6,1; 6,2;
6,3;
6,4; 6,5. You ALSO lose two shields with a 6,6. Out of 36 possible dice
combinations, 11 of them result in at least one shield dropping. (Or, there
is a 5/6 chance of not losing a shield in a level 1 ship, and a 5/6 *
5/6
chance of not losing a shield in a level 2 ship.)

It's even more pronounced with 3 shields. There are 216 combinations. There
are 91 shield dropping combinations out of 216 (if I've done the math
correctly) for three shields, or a 42.13% chance that NO SHIELD will be
dropped when checking three shields. You lose 2 shields in 15/216
combinations, and three shields in 1/216 combinations.

So the loses are:
no reduction:    5/6 for L1 ship, 25/36 for L2 ship, 125/216 for L3 ship
lose at least 1 level: 1/6, 12/36, 91/216
lose at least 2 levels: 0/6, 1/36, 16/216
lose 3 levels: 0/6, 0/6, 1/216

This error gets bigger at the second threshold. For ships with 2 shields,
there are 16 combinations of 36 that result in the loss of one shield, and 4
that result in the loss of two shields. This means that there is a 16/36
chance that a level 2 ship will not lose a shield. For level 3 shields, the
chance of not losing a shield is 64/216.

Unfortunately, this error makes the rest of your analysis inaccurate.

> On 8 Nov 1996, Brian Bell wrote:

> Below are some stats that take into account losses to threshold rolls.
I have
> selected a Privateer (18 tons), Escort Cruiser (26 tons), and

No, not correct. If you have 1 shield, the chance to keep it after the first
treshold check is 83.3333333....%, yes. If you have 2 shields, the
chance to have both of them after the first check is (5/6)*(5/6) = 25/36

= 69.44444...%, the probability to lose both is (1/6)*(1/6) = 1/36 =
2.7777...%, and the chance to lose one but not the other is
1 - 25/36 - 1/36 = 10/36 = 27.777777...%.

The correct chances to lose/keep your shields are (repost) (FRN):

		    Treshold check number: 0	 1     2     3
Origingal level-1:
Level-1                                   100%  83%   56%   28%
Level-0                                    0%   17%   44%   72%

Original level-2:
Level-2                                   100%  69%   31%    8%
Level-1                                    0%   28%   54%   42%
Level-0                                    0%    3%   15%   50%

Original level-3:
Level-3                                   100%  58%   17%    2%
Level-2                                    0%   35%   41%   17%
Level-1                                    0%    7%   33%   43%
Level-0                                    0%   0.5%   9%   38%

However, the chance that a ship which has already lost a shield in one
treshold check will lose another one is not the same as the chance for a

ship which didn't lose shields in the first check to lose them... so I'm

not sure if I can simply take 'average' shield levels to determine how much
damage a ship can absorb. I don't think I can, but I haven't done enough
thinking on this yet.

Regards,

-------------------- Begin Original Message --------------------
Message text written by Allan Goodall:

"It's even more pronounced with 3 shields. There are 216 combinations. There
are 91 shield dropping combinations out of 216 (if I've done the math
correctly) for three shields, or a 42.13% chance that NO SHIELD will be
dropped when checking three shields. You lose 2 shields in 15/216
combinations, and three shields in 1/216 combinations."
-------------------- End Original Message --------------------
57.87% of NO SCREEN loss. It would be 42.13% chance that at least one screen
would be dropped, 6.94% of loosing 2 screens, and 0.46% that 3 screens would
be lost. The 42.13% chance of loosing 1 screen contains all 6.94% chance of
loosing 2 screens and the 0.46% chance of loosing 3 screens.

But still, my predictions were way off. ;-)