• 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.

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