填充第 9 題：
某人初次投籃投進的機率為\(\displaystyle \frac{1}{2}\)，當他投進一球後，則下一球投進的機率為\(\displaystyle \frac{4}{5}\)，當他有一球投不進後，下一球投進的機率為\(\displaystyle \frac{3}{5}\)。若某人總共投四次球，則投進次數的期望值為？

初始矩陣 \(\displaystyle X_1=\left[\begin{array}{cc}\displaystyle \frac{1}{2}\\ \frac{1}{2}\end{array}\right]\)

轉移矩陣 \(\displaystyle P=\left[\begin{array}{cc}\displaystyle \frac{4}{5}&\frac{3}{5}\\ \frac{1}{5}&\frac{2}{5}\end{array}\right]\)

\(\displaystyle X_2=PX_1=\left[\begin{array}{cc}\displaystyle \frac{7}{10}\\ \frac{3}{10}\end{array}\right]\)

\(\displaystyle X_3=PX_2=\left[\begin{array}{cc}\displaystyle \frac{37}{50}\\ \frac{13}{50}\end{array}\right]\)

\(\displaystyle X_4=PX_3=\left[\begin{array}{cc}\displaystyle \frac{187}{250}\\ \frac{63}{250}\end{array}\right]\)

所求＝\(\displaystyle 1\cdot\frac{1}{2}+1\cdot\frac{7}{10}+1\cdot\frac{37}{50} +1\cdot\frac{187}{250}=\frac{336}{125}\)

填充第 9 題說明：由於每次只投一球，所以在第 \(n\) 次投球會進球的機率，就是在第 \(n\) 次投球進球球數的期望值。