| $X$ | 1 | 2 | 3 | Total |
| $\mathrm { P } ( X )$ | 0.5 | 0.3 | 0.2 | 1 |
| $\bar { X }$ | 1 | 1.5 | 2 | 2.5 | 3 |
| Frequency | 1 | $a$ | $b$ | 2 | 1 |
| $\mathrm { P } ( \bar { X } )$ | 0.25 | $c$ | $d$ | 0.12 | 0.04 |
The following is a probability distribution table of a certain population.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$X$ & 1 & 2 & 3 & Total \\
\hline
$\mathrm { P } ( X )$ & 0.5 & 0.3 & 0.2 & 1 \\
\hline
\end{tabular}
\end{center}
When a sample of size 2 is drawn with replacement from this population, the probability distribution table of the sample mean $\bar { X }$ is as follows.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$\bar { X }$ & 1 & 1.5 & 2 & 2.5 & 3 \\
\hline
Frequency & 1 & $a$ & $b$ & 2 & 1 \\
\hline
$\mathrm { P } ( \bar { X } )$ & 0.25 & $c$ & $d$ & 0.12 & 0.04 \\
\hline
\end{tabular}
\end{center}
Find the value of $100 ( b + c )$. [4 points]