While this equation is not very illuminating, it does allow us to efficiently compute the distribution and thus answer various questions regarding the results of dice rolls.
@ -63,6 +65,53 @@
\caption{Example of candidate combinations of $f$ and $c_f$ with upper and lower bounds for $c_f$ shown. In this plot, $d=12$ and $m=8$.\label{fig:candidates}}
\end{figure}
\subsection{Probability of Success}
\begin{multicols}{2}
The first question to ask is: ``What is the probability of success, given a fixed \stat{DT}?''
We will use the binomial distribution $\text{B}(20,7/20)$ for the size of the dice pool, to compute the expectation. This is a fairly arbitrary choice that assumes 7 is the average pool size.
\Cref{tab:cos} shows the probability of success for a given \stat{DT} as well as number of successes in a number of attempts.
\end{multicols}
\begin{table}[htb]
\centering
\begin{tabular}{r | r r c l}
\stat{DT}&$p$&\multicolumn{3}{c}{Succeeds}\\
1 & 98\%& 51 & in & 52 \\
2 & 95\%& 20 & in & 21 \\
3 & 90\%& 8 & in & 9 \\
4 & 82\%& 4 & in & 5 \\
5 & 70\%& 2 & in & 3 \\
6 & 56\%& 1 & in & 2 \\
7 & 42\%& 2 & in & 5 \\
8 & 28\%& 1 & in & 4 \\
9 & 17\%& 1 & in & 6 \\
10 & 9\%& 1 & in & 12 \\
11 & 4\%& 1 & in & 26 \\
\end{tabular}
\caption{Expected chance of success.\label{tab:cos}}
% Succeeds & 51 in 52 & 20 in 21 & 8 in 9 & 4 in 5 & 2 in 3 & 1 in 2 & 2 in 5 & 1 in 4 & 1 in 6 & 1 in 12 & 1 in 26
%\end{tabular}
%\caption{Expected chance of success.\label{tab:cos2}}
%\end{figure}
\subsection{Expected Margin of Success}
\begin{multicols}{2}
We also want to know what the expected margin of success is, given that the player succeeded in the roll, $\text{E}\left[M | M \geq0\right]$. \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.