Why I hate SBR!!!

it is well known that the expected value for damage on an SBR is 3.5 IPCs (given the bomber survives AA fire and given the territory is 6 IPCs or more).  there are a few posts taking a stab at the distribution for heavy bombers in LHTR, but no one has formulated it correctly.  the work below shows the average SBR damage is 5.472 IPCs per heavy bomber.

in LHTR, heavy bombers roll 2 dice, take the max, add one, limit to the territory value, and that’s the IPC damage inflicted.

let A and B be independent random variables corresponding to the two dice rolled.  they are distributed as:
P(A=a) ={ 1/6  a=1,2,3,4,5,6
        { 0    otherwise

P(B=b) ={ 1/6  b=1,2,3,4,5,6
        { 0    otherwise

let C be the derived random variable defined as:
C = max(A,B) + 1

Theorem: C is distributed as:
P(C=c) ={  1/36    c=2
        {  3/36    c=3
        {  5/36    c=4
        {  7/36    c=5
        {  9/36    c=6
        { 11/36    c=7
        { 0        otherwise

Proof:
note: <= is the less-than-or-equal-to operator
note: E is the summation operator
P( C<=c ) = P( max(A,B)+1 <= c )

6
      = E  P( max(A,b)+1 <= c | B=b ) P(B=b)
          b=1

6
      = E  P( max(A,b) <= c-1 ) P(B=b)
          b=1

c-1
      = 1/6 * E  P( A <= c-1 )      c<=7
                b=1

c-1 c-1
      = 1/6 * E  E  P(A=a)          c<=7
                b=1 a=1

c-1 c-1
      = 1/36 * E  E  1              c<=7
                  b=1 a=1

= 1/36 * (c-1)^2              c<=7

and
P(C=c) = P(C<=c) - P(C<=c-1)
      ={  1/36    c=2
        {  3/36    c=3
        {  5/36    c=4
        {  7/36    c=5
        {  9/36    c=6
        { 11/36    c=7
        { 0        otherwise

Analysis:
note: E is the expectation operator
so, if one were heavy SBR’ing a 7 (or more) territory, the expected damage is:
E© = 21/36 + 33/36 + 45/36 + 57/36 + 69/36 + 711/36 = 197/36 = 5.472 IPC

so is the extra ~2 IPC per bomber worth the investment?  i’d like to know what other analysts think now that i’ve quantified the difference.

Oh, I didn’t realize that.

Thanks for the link. I’ll talk to my buddies about adopting those rules for our next game.

I’ll try this out… sounds like fun

Update:

Germany holds most of Africa with 100% of Japanese forces being diverted to Egypt.  England’s putting 6 units a round into Libya (3 infantry, 3 armor) and hitting Germany for 7 IPC in damage every round (taking 5 damage in return.)  America is invading through Norway and W. Europe with 10 units a round with 5 bombers doing 17/18 IPC in damage (alternate rounds) and taking 12/13 IPC in damage (leaving me with plenty of income to keep at least 10 infantry, usually 6 infantry, 3 artillery, 1 armor) going into Asia.

Germany has yet to see a round after 1 with more then 40 IPC income.  They are currently averaging closer to 38 IPC - 25 in SBRs for a net of 13 IPC to Russia’s 20-28 IPCs.  Japan is no threat as they have to move all builds into Egypt to counter English reinforcements.

Germany has no Atlantic fleet.  Their med fleet is augmented with Japanese battleships to prevent American fighters from sinking them.

This is from page twenty of LHTR pdf file

Strategic Bombing Raids: a bomber can make an economic attack against an enemy industrial
complex. During the opening fire step of combat, bombers that survive any antiaircraft
fire may attack the industrial complex. Roll one die for each surviving bomber. The
opponent must surrender that many IPCs to the bank. The maximum combined damage inflicted in one turn by all strategic bombing raids on one industrial complex and any rocket attack on the same industrial complex is the territory’s income value.

Italics mine.

This from the out of box rules page 28:
Special Abilities
Strategic Bombing Raids: A bomber can make an economic attack against an enemy industrial complex.
During the opening fire step of combat, bombers that survive any antiaircraft fire may attack the industrial complex. Roll one die for each surviving bomber (the maximum each bomber can inflict is the territory’s income value). The opponent must surrender that many IPCs to the bank.
Weapons Development—Heavy Bombers: If you have the Heavy Bombers development, all your
bombers roll two dice on an attack instead of one. That means each bomber can score 2 hits in a regular attack, or inflict two dice of IPC loss in a strategic bombing raid. (It still cannot inflict more loss than the territory’s income value.)

Italics and bold mine.

If you are playing out of the box SBRs can do some hurt. Not as much hurt in “tourney” rules. Those are the out of box rules pdf from the Avalon Hill website. I think those are the latest LHTRs but it is from the link to the pdf version in the sticky thread. I guess that leaves the question of if in LHTR by turn they mean UK’s move or R thru US moves. My earlier post while not quite wrong is misprecise.

@ncscswitch:

@Jennifer:

My bad…I thought I was in the classic thread.  Much apologies.

Not a problem, have done that myself recently :-)

Yea, it’s easy to do when you hit “show unread posts since last visit” instead of hunting by catagory. wink

Because in WWII when bombers attack, the fighters deffending the territories and industrial complex……

So I think when the Attacker use Bomber Raid sent Bombers and Fighters, and defender could be use the defender fighters before the dice roller to the Industrial Complex