\( 設P_n表示取n次後，A箱中為一黑一白的機率 \)
\(
推得 \displaystyle P_n = \frac{3}{4}P_{n - 1} + (1 - P_{n - 1} ) \times \frac{1}{2}，整理可得
\)
\(
\left\{ \begin{array}{l}
\displaystyle P_1 = \frac{3}{4} \\
\displaystyle P_n = \frac{1}{4}P_{n - 1} + \frac{1}{2},n \ge 2 \\
\end{array} \right.
\)
\(
\displaystyle 由遞迴式依序代入得 P_2 = \frac{{11}}{{16}},P_3 = \frac{{43}}{{64}}
\)
