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HeNine 2 years ago
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dice.tex
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\begin{multicols}{2}
We also want to know what the expected margin of success is, given that the player succeeded in the roll, $\text{E}\left[M | M \geq 0\right]$. \Cref{tab:exp_mos} shows the expectations.
Of note are the expected numbers of successes for \stat{DT} 11. The requirement for critical successes there causes a large variance in results.
Of note are the expected numbers of successes for \stat{DT} 11. The requirement for critical successes there causes a large variance in results. It is also important to remember that this does not show the full range of results. The highest possible result is $2d$, which for one die is the only possible (successful) result, while other numbers of dice have wider ranges.
\end{multicols}
\begin{figure}[htb]
\begin{table}[htb]
\centering
\begin{tabular}{r | c c c c c c c c}
& \multicolumn{8}{c}{Difficulty Threshold} \\
d & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \cline{2-9}
1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\
2 & 2 & 2 & 1 & 1 & 1 & 1 & 0 & 0 \\
3 & 3 & 3 & 2 & 2 & 1 & 1 & 1 & 1 \\
4 & 4 & 3 & 3 & 2 & 2 & 1 & 1 & 0 \\
5 & 5 & 4 & 3 & 3 & 2 & 1 & 1 & 1 \\
6 & 6 & 5 & 4 & 3 & 2 & 2 & 1 & 1 \\
7 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 1 \\
8 & 8 & 7 & 5 & 4 & 3 & 2 & 1 & 1 \\
9 & 9 & 8 & 6 & 5 & 3 & 2 & 1 & 1 \\
10 & 10 & 8 & 7 & 5 & 4 & 2 & 1 & 1 \\
\begin{tabular}{r | S[table-format=2.1] S[table-format=2.1] S[table-format=2.1] S[table-format=2.1] S[table-format=2.1] S[table-format=2.1] S[table-format=2.1] S[table-format=2.1] S[table-format=2.1] S[table-format=2.1] S[table-format=2.1] S[table-format=2.1]}
& \multicolumn{10}{c}{Difficulty Threshold} \\
$d$ & \multicolumn{1}{c}{1} & \multicolumn{1}{c}{2} & \multicolumn{1}{c}{3} & \multicolumn{1}{c}{4} & \multicolumn{1}{c}{5} & \multicolumn{1}{c}{6} & \multicolumn{1}{c}{7} & \multicolumn{1}{c}{8} & \multicolumn{1}{c}{9} & \multicolumn{1}{c}{10} & \multicolumn{1}{c}{11} \\ \cline{2-12}
1 & 1.1 & 1.1 & 1.1 & 1.1 & 1.1 & 1.2 & 1.2 & 1.2 & 1.3 & 1.5 & 2.0 \\
2 & 2.1 & 1.8 & 1.5 & 1.3 & 1.1 & 1.0 & 0.8 & 0.7 & 0.7 & 0.7 & 1.0 \\
3 & 2.6 & 2.3 & 2.1 & 1.9 & 1.8 & 1.6 & 1.5 & 1.3 & 1.1 & 0.7 & 0.3 \\
4 & 3.5 & 3.1 & 2.7 & 2.3 & 1.9 & 1.6 & 1.4 & 1.2 & 1.1 & 1.2 & 2.0 \\
5 & 4.4 & 3.7 & 3.1 & 2.7 & 2.3 & 2.0 & 1.7 & 1.4 & 1.1 & 0.9 & 1.1 \\
6 & 5.1 & 4.3 & 3.6 & 3.0 & 2.5 & 2.1 & 1.7 & 1.5 & 1.3 & 1.1 & 0.5 \\
7 & 6.0 & 5.0 & 4.1 & 3.4 & 2.8 & 2.3 & 2.0 & 1.6 & 1.3 & 1.2 & 1.9 \\
8 & 6.8 & 5.6 & 4.6 & 3.7 & 3.0 & 2.5 & 2.0 & 1.7 & 1.4 & 1.1 & 1.0 \\
9 & 7.5 & 6.2 & 5.1 & 4.1 & 3.3 & 2.7 & 2.2 & 1.8 & 1.4 & 1.2 & 0.6 \\
10 & 8.4 & 6.9 & 5.5 & 4.4 & 3.5 & 2.8 & 2.2 & 1.8 & 1.5 & 1.2 & 1.8 \\
11 & 9.2 & 7.5 & 6.0 & 4.8 & 3.7 & 3.0 & 2.4 & 1.9 & 1.5 & 1.2 & 1.0 \\
12 & 10.0 & 8.2 & 6.5 & 5.1 & 4.0 & 3.1 & 2.4 & 1.9 & 1.6 & 1.3 & 0.7
\end{tabular}
\caption{Expected number of successes.\label{tab:exp_mos}}
\end{figure}
\end{table}
\section{Combat Pool Steady State}

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