probability of success

dice.tex

@@ -1,6 +1,8 @@
 \chapter{Dice}
 
-\section{Distribution of Successes}
+\section{Basics of Dice}
+
+\subsection{Distribution of Successes}
 
 \begin{multicols}{2}
     For further analysis it will be useful to have a distribution function for margins of success.
@@ -48,9 +50,9 @@
     Putting the sum together, and adding floors and ceilings to limit the sum to integers, we get:
     \begin{align}
         Pr(M=m) & = \sum_{f=0}^{\left\lfloor \frac{2d-m}{3} \right\rfloor}\sum_{c_f=\max(0,\left\lceil\frac{m-d+2f}{-3}\right\rceil)}^{\left\lfloor\frac{-m+2d-3f}{4}\right\rfloor} \\
-                &\quad\; \frac{d!}{c_s!s!f!c_f!}
-        \frac{n_s^s n_f^f}{12^{d}} \\
-        n_f     & = \stat{dt}-1 \\
+                & \quad\; \frac{d!}{c_s!s!f!c_f!}
+        \frac{n_s^s n_f^f}{12^{d}}                                                                                                                                                  \\
+        n_f     & = \stat{dt}-1                                                                                                                                                     \\
         n_s     & = 12-n_s
     \end{align}
     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}}
+\end{table}
+
+% \begin{figure}[htb]
+%     \centering
+%     \begin{tabular}{r c c c c c c c c c c c}
+%         \stat{DT} & 1        & 2        & 3      & 4      & 5      & 6      & 7      & 8      & 9      & 10      & 11        \\ \hline
+%         $p$       & 98\%     & 95\%     & 90\%   & 82\%   & 70\%   & 56\%   & 42\%   & 28\%   & 17\%   & 9\%     & 4\%       \\
+%         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, $\text{E}\left[M | M \geq 0\right]$. \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.
+\end{multicols}
+
 \begin{figure}[htb]
     \centering
     \begin{tabular}{r | c c c c c c c c}