Q50

A message is made up entirely of characters from the set $X=\{P, Q, R, S, T\}$. The table of probabilities for each of the characters is shown below:

$$\begin{array}{|c|c|}\hline \textbf{Character} & \textbf{Probability } \\\hline \text{$P$} & \text{$0.22$} \\ \text{$Q$} & \text{$0.34$} \\ \text{$R$} & \text{$0.17$} \\ \text{$S$} & \text{$0.19$} \\ \text{$T$} & \text{$0.08$} \\ \hline\text{Total} & \text{$1.00$} \\\hline \end{array}$$

If a message of $100$ characters over $X$ is encoded using Huffman coding, then the expected length of the encoded message in bits is ______.