probability of success

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HeNine 2 years ago
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\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.
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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.
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\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}

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