Lets define all:\\
A is \# of eve accepted w/o Endcap ~~~(1)\\
B is \# of eve rejected if Endcap is used\\
C is \# of eve accepted if Endcap is used\\
X is estimator of \# of eve accepted if East \& West Endcap were used.\\
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Obviously B+C=A ; B \& C are statistically independent,\\ so
~~$\sigma(A)=\sqrt{B+C}$; ~~$\sigma(B)=\sqrt{B}$; ~~$\sigma(C)=\sqrt{C}$ ~~~(2)\\
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The estimator X is approximated as\\
$X=C-B$ ~~~(3)\\
Lets define the signal/background ratio $R_{S/B}$, where as background we count all QCD events not discarded if neither Endcap is not used \\
$R_{S/B}=\frac{X}{A-X}$ ~~~(4) \\
Since numerator \& denominator are correlated (via X -variable) lets do some substitutions:\\
$R_{S/B}=\frac{C}{2B} - \frac{1}{2}$ ~~~(5)\\
Now we have ratio of statistically independent variables and it is easy to compute: \\
~~ $\sigma(R_{S/B})=\frac{C-B}{2B}\sqrt{\frac{1}{C} + \frac{1}{B}}$ ~~~(6)
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