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—空军单位被防空武器(AA)击落的概率计算方法—

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发表于 2004-2-4 11:33:28 | 显示全部楼层 |阅读模式
本文摘译自civfanatics


注:1、只有处于被空军单位攻击的方格中的防空单位才会开火,相邻的方格即使也有防空单位也不会开火。
    2、防空单位的顺序是按照防空能力从高到低排定的,不能改变。(原句是Note: AAUnits must be sorted in decending order, with highest Defense AA unit as AAUnit#1)
——————————————————————————————————————
公式说明:

AA#1、AA#2、AA#3……表示防空单位防守顺序编号
AirD=1、2、3……表示空军单位的防御能力(比如:Bomber轰炸机=2,Stealth Bomber=5)
AAD#1=1、2、3……表示第1个防空单位的防空能力,AAD#2=1、2、3……表示第2个防空单位的防空能力(比如:Flak高射炮=2,SAM防空导弹=4)
防空单位防守失败的概率=1-击落敌机的概率
——————————————————————————————————————
空军单位被第1个防空单位(AA#1)击落的概率=1-[AirD/(AirD+(AAD#1x0.1))]

空军单位被第2个防空单位(AA#2)击落的概率={1-[AirD/(AirD+(AAD#2x0.1))]}x第1个防空单位防守失败的概率

空军单位被第3个防空单位(AA#3)击落的概率={1-[AirD/(AirD+(AAD#3x0.1))]}x第1个防空单位防守失败的概率x第2个防空单位防守失败的概率

空军单位被第4个防空单位(AA#4)击落的概率={1-[AirD/(AirD+(AAD#4x0.1))]}x第1个防空单位防守失败的概率x第2个防空单位防守失败的概率x第3个防空单位防守失败的概率

…………依次类推

*最终这个方格对空军拦截成功的概率=所有防空单位拦截成功概率的总和
——————————————————————————————————————
附:几个例子

1、一架轰炸机(AirD=2)攻击有1个Flak高射炮(AAD=2)驻守的方格:

轰炸机被Flak高射炮击落的概率=1-[2/(2+(2x0.1))]=0.0909=9.09%
Flak高射炮防守失败的概率=1-9.09%=90.91%

这个方格对空军拦截成功的概率=9.09%
——————————————————————————————————————
2、一架轰炸机(AirD=2)攻击有2个Flak高射炮(AAD=2)驻守的方格:

轰炸机被第1个Flak高射炮(AA#1)击落的概率=1-[2/(2+(2x0.1))]=0.0909=9.09%
第1个Flak高射炮防守失败的概率=1-9.09%=90.91%

轰炸机被第2个Flak高射炮(AA#2)击落的概率=1-[2/(2+(2x0.1))]=0.0909=9.09%
第2个Flak高射炮防守失败的概率=1-9.09%=90.91%

这个方格对空军拦截成功的概率=9.09%+(9.09%x90.91%)=9.09%+8.26%=17.35%
——————————————————————————————————————
3、一架轰炸机(AirD=2)攻击有1个SAM导弹(AAD=4)和1个Flak高射炮(AAD=2)驻守的方格:

轰炸机被SAM导弹(AA#1)击落的概率=1-[2/(2+(4x0.1))]=0.1667=16.67%
SAM导弹防守失败的概率=1-16.67%=83.33%

轰炸机被Flak高射炮(AA#2)击落的概率=1-[2/(2+(2x0.1))]=0.0909=9.09%
Flak高射炮防守失败的概率=1-9.09%=90.91%

这个方格对空军拦截成功的概率=16.67%+(9.09%x83.33%)=24.24%
——————————————————————————————————————
4、一架轰炸机(AirD=2)攻击有2个SAM导弹(AAD=4)驻守的方格:

轰炸机被第1个SAM导弹(AA#1)击落的概率=1-[2/(2+(4x0.1))]=0.1667=16.67%
第1个SAM导弹防守失败的概率=1-16.67%=83.33%

轰炸机被第2个SAM导弹(AA#2)击落的概率=1-[2/(2+(4x0.1))]=0.1667=16.67%
第2个SAM导弹防守失败的概率=1-16.67%=83.33%

这个方格对空军拦截成功的概率=16.67%+(16.67%x83.33%)=30.56%
——————————————————————————————————————
5、一架隐形轰炸机(AirD=5)攻击有1个SAM导弹(AAD=4)驻守的方格:

隐形轰炸机被SAM导弹击落的概率=1-[5/(5+(4x0.1))]=0.0741=7.41%
SAM导弹防守失败的概率=1-7.41%=92.59%

这个方格对空军拦截成功的概率=7.41%
——————————————————————————————————————
6、一架隐形轰炸机(AirD=5)攻击有1艘AEGIS宙斯盾巡洋舰(AAD=3)驻守的方格:

隐形轰炸机被AEGIS宙斯盾巡洋舰击落的概率=1-[5/(5+(3x0.1))]=0.0566=5.66%
AEGIS宙斯盾巡洋舰防守失败的概率=1-5.66%=94.34%

这个方格对空军拦截成功的概率=5.66%
——————————————————————————————————————


由此可见,只有用大批的防空才能有效的拦截空军,而面对空军相当脆弱的海军在行进时一定要在同一方格内!

[ 本帖最后由 96nn 于 2006-7-24 08:55 PM 编辑 ]
 楼主| 发表于 2004-2-4 11:40:57 | 显示全部楼层
原文如下:


Here is the interception formula...

Chance Air Unit is Destoyed By AAUnit #1 = 1 - [AirD/(AirD + (AAUnitD#1 * 0.1))]

Chance Air Unit is Destoyed By AAUnit #2 = {1 - [AirD/(AirD + (AAUnitD#2 * 0.1))]} * Chance AAUnit#1 Misses

Chance Air Unit is Destoyed By AAUnit #3 = {1 - [AirD/(AirD + (AAUnitD#3 * 0.1))]} * Chance AAUnit#1 Misses * Chance AAUnit#2 Misses

Chance Air Unit is Destoyed By AAUnit #4 = {1 - [AirD/(AirD + (AAUnitD#4 * 0.1))]} * Chance AAUnit#1 Misses * Chance AAUnit#2 Misses * Chance AAUnit#3 Misses

Interception Percentage = Sum of above fomulas

It is done as above because AAUnit#2 only gets a chance to fire if AAUnit#1 Misses... AAunit#3 only gets a chance if #1 and #2 miss, etc...

Luckily, these formula's approximate to:

1 - [{AirD/(AirD + (AverageOfAAUnit#1-#4D * 0.1))}^#AAUnits]

This is not an exact match, but is close enough for government work...  Errors only start to show when the Average AA Unit Value is very different than either end (AAUnit#1D and the MinDefender)... For 'dead nuts'  calculations , use the long formulas...

Note: AAUnits must be sorted in decending order, with highest Defense AA unit as AAUnit#1
---------------------------------------------------------------------
TEST #1 - #3
A Bomber Attacks a tile with 1 Flak
AirD = 2

AirUnit Destroyed By AAUnit#1 = 1 - [2/(2 + (2 * 0.1))] = 0.0909
AAUnit#1 Missed AirUnit = 1 - 0.0909 = 0.9090

Percentage Air Unit Destroyed = 0.0909 (9.09%)

Test #1-#3: 421/4500 Destroyed = 9.35%
Standard Deviation = 19.3 Bombers (+/- 0.43%)
1 s.d. = 9.09% +/- 0.43% (8.66% - 9.52%) <---
---------------------------------------------------------------------
TEST #4
A Bomber Attacks a tile with 2 Flak...
AirD = 2

AirUnit Destroyed By AAUnit#1 = 1 - [2/(2 + (2 * 0.1))] = 0.0909
AAUnit#1 Missed AirUnit = 1 - 0.0909 = 0.9090

AirUnit Destroyed By AAUnit#2 = 1 - [2/(2 + (2 * 0.1))] = 0.0909
AAUnit#2 Missed AirUnit = 1 - 0.0909 = 0.9090

Percentage Air Unit Destroyed = 0.0909 + (0.0909 * 0.9090) = 0.0909 + 0.0826 = 0.1735 (17.35%)

Test #4: 88/500 Destroyed = 17.6%
Standard Deviation = 8.52 Bombers (+/- 1.70%)
1 s.d. = 17.6% +/- 1.70% (15.9% - 19.3%) <---
--------------------------------------------------------------------
TEST #6
A Bomber Attacks a tile with 1 MSAM and a Flak...
AirD = 2

AirUnit Destroyed By AAUnit#1 = 1 - [2/(2 + (4 * 0.1))] = 0.1667
AAUnit#1 Missed AirUnit = 1 - 0.1667 = 0.8333

AirUnit Destroyed By AAUnit#2 = 1 - [2/(2 + (2 * 0.1))] = 0.0909
AAUnit#2 Missed AirUnit = 1 - 0.0909 = 0.9090

Percentage Air Unit Destroyed = 0.1667 + (0.0909 * 0.8333) = 0.1667 + 0.0757 = 0.2424 (24.24%)

Test #6: 114/500 Destroyed = 22.8%
Standard Deviation = 9.38 Bombers (+/- 1.88%)
1 s.d. = 22.80% +/- 1.88% (20.92% - 24.68%) <---
---------------------------------------------------------------------
TEST #9
A Bomber Attacks a tile with 2 MSAMs and a Flak...
AirD = 2

AirUnit Destroyed By AAUnit#1 = 1 - [2/(2 + (4 * 0.1))] = 0.1667
AAUnit#1 Missed AirUnit = 1 - 0.1667 = 0.8333

AirUnit Destroyed By AAUnit#2 = 1 - [2/(2 + (4 * 0.1))] = 0.1667
AAUnit#2 Missed AirUnit = 1 - 0.1667 = 0.8333

AirUnit Destroyed By AAUnit#3 = 1 - [2/(2 + (2 * 0.1))] = 0.0909
AAUnit#3 Missed AirUnit = 1 - 0.0909 = 0.9090

Percentage Air Unit Destroyed = 0.1667 + (0.1667 * 0.8333) + (0.0909 * 0.8333 * 0.8333) = 0.1667 + 0.1389 + 0.0631 = 0.3687 (36.87%)

Test #9: 156/500 Destroyed = 31.2%
Standard Deviation = 10.79 Bombers (+/- 2.16%)
1 s.d. = 36.87% +/- 2.16% (34.71% - 39.03%)
2 s.d. = 36.87% +/- 4.32% (32.55% - 41.19%)
3 s.d. = 36.87% +/- 6.48% (30.39% - 43.35%) <--- Very odd result!
---------------------------------------------------------------------
TEST #10
A Bomber Attacks a tile with 4 MSAMs
AirD = 2

AirUnit Destroyed By AAUnit#1 = 1 - [2/(2 + (4 * 0.1))] = 0.1667
AAUnit#1 Missed AirUnit = 1 - 0.1667 = 0.8333

AirUnit Destroyed By AAUnit#2 = 1 - [2/(2 + (4 * 0.1))] = 0.1667
AAUnit#2 Missed AirUnit = 1 - 0.1667 = 0.8333

AirUnit Destroyed By AAUnit#3 = 1 - [2/(2 + (4 * 0.1))] = 0.1667
AAUnit#3 Missed AirUnit = 1 - 0.1667 = 0.8333

AirUnit Destroyed By AAUnit#4 = 1 - [2/(2 + (4 * 0.1))] = 0.1667
AAUnit#4 Missed AirUnit = 1 - 0.1667 = 0.8333

Percentage Air Unit Destroyed = 0.1667 + (0.1667 * 0.8333) + (0.1667 * 0.8333 * 0.8333) +

(0.1667 * 0.8333 * 0.8333 * 0.8333) = 0.1667 + 0.1389 + 0.1158 + 0.0965 = 0.5179 (51.79%)

Test #10: 270/500 Destroyed = 54.0%
Standard Deviation = 11.17 Bombers (+/- 2.23%)
1 s.d. = 51.79% +/- 2.23% (49.56% - 54.02%) <---
---------------------------------------------------------------------
A Stealth Bomber Attacks a tile with 2 AEGIS Cruisers, 1 Destroyer and 1 Battleship...
AirD = 5

AirUnit Destroyed By AAUnit#1 = 1 - [5/(5 + (3 * 0.1))] = 0.0566
AAUnit#1 Missed AirUnit = 1 - 0.0566 = 0.9434

AirUnit Destroyed By AAUnit#2 = 1 - [5/(5 + (3 * 0.1))] = 0.0566
AAUnit#2 Missed AirUnit = 1 - 0.0566 = 0.9434

AirUnit Destroyed By AAUnit#3 = 1 - [5/(5 + (2 * 0.1))] = 0.0385
AAUnit#3 Missed AirUnit = 1 - 0.0385 = 0.9615

AirUnit Destroyed By AAUnit#4 = 1 - [5/(5 + (1 * 0.1))] = 0.0196
AAUnit#4 Missed AirUnit = 1 - 0.0196 = 0.9804

Percentage Air Unit Destroyed = 0.0566 + (0.0566 * 0.9434) + (0.0385 * 0.9434 * 0.9434) + (0.0196 * 0.9434 * 0.9434 * 0.9804) = 0.0566 + 0.0534 + 0.0343 + 0.0171 = 0.1614 (16.14%)
---------------------------------------------------------------------

[ 本帖最后由 96nn 于 2006-7-24 08:58 PM 编辑 ]
发表于 2004-2-4 11:52:40 | 显示全部楼层
hao fu  zai     ~
发表于 2004-2-4 11:56:08 | 显示全部楼层
突出了制空权的重要性
发表于 2004-2-4 12:25:40 | 显示全部楼层
请问战机的拦截半径是怎么算的?拦截半径等于作战半径么?战机拦截概率怎么算?
 楼主| 发表于 2004-2-5 14:28:23 | 显示全部楼层
以下是引用miaohy在2004-2-4 12:25:40的发言:
请问战机的拦截半径是怎么算的?拦截半径等于作战半径么?战机拦截概率怎么算?

这个偶没有研究~也没有看到过这方面的文章,等知道了再告诉你吧
发表于 2004-6-12 23:51:51 | 显示全部楼层
顶出来先``
发表于 2004-6-28 15:46:14 | 显示全部楼层
a
发表于 2004-6-30 10:04:57 | 显示全部楼层
做一条这么长的应用题才算出被击落的概率!
发表于 2004-10-3 22:40:52 | 显示全部楼层
似的发射点法
发表于 2004-12-10 08:12:44 | 显示全部楼层
按照上面的公式,防空单位的排列顺序是起较大作用的,那防空单位的顺序是怎样排列的呢?
发表于 2006-1-29 03:27:45 | 显示全部楼层
請問在文明3主程式裡,空軍單位都有顯示移動數值,如轟炸機0(8).2.1/1
是指攻擊為0,轟炸能力是8,防禦是2,1/1是行動次數,如果轟炸機執行完任務,
則顯示0(8).2.0/1,0/1表示執行完畢.
但在征服裡,卻都沒有顯示,如轟炸機0(8).2
這樣一來,城市裡面有10多架轟炸機的話,一看傻眼,根本不知道哪一架已經執行完任務.各位是如何指揮空軍作戰的呢??
发表于 2006-1-29 10:06:14 | 显示全部楼层

回复:(Jibuli)—空军单位被防空武器(AA)击落的概...

結論是用防空武器還是有最佳數量的,並不是越多,防空能力就越高,而且不可能超過100%.
因為第1個防空武器沒攔截成功的話,後面的攔截成功率越來越低,成一個收斂級數,會趨於某定值.
发表于 2009-2-2 12:47:39 | 显示全部楼层
楼主计算的真是精确啊,我更愿意打陆战。
发表于 2009-2-3 07:15:06 | 显示全部楼层
五年前的贴又被挖上来了
明天此帖五周年纪念,大家来观摩
发表于 2009-6-16 17:22:29 | 显示全部楼层
有没有帖子讲空战击落概率的啊?
发表于 2009-7-17 17:06:03 | 显示全部楼层
我只想知道为什么我的喷气战斗机不拦截轰炸机的?
发表于 2009-7-17 17:18:36 | 显示全部楼层
楼上应该是几率问题。还有,战斗机被命令执行空优任务的第一回合,因为机动力耗尽,所以不能拦截。
发表于 2009-7-20 00:03:50 | 显示全部楼层
原帖由 96nn 于 2009-7-17 17:18 发表
楼上应该是几率问题。还有,战斗机被命令执行空优任务的第一回合,因为机动力耗尽,所以不能拦截。

       但概率也未免太小了吧,我玩了那么久的文明3,从没见过我的喷气战斗机拦截对方的飞机,但我轰炸电脑的国家时又总是被它们拦截到。我的是天人互动1.07f的。
发表于 2009-7-20 12:13:42 | 显示全部楼层
因为文明3有一个隐藏设定,或者说是AI的作弊,就是当玩家的战斗机执行空优任务时,AI似乎倾向于避免发动空袭,但也不尽然,确实是概率问题。

我就有一次,征服1.22,世界地图,在环球多个城市共有30~40架喷气战斗机在值勤,下个回合俄国宣我,他20多架轰炸机中有一多半飞过来,被我拦截了10多架。
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