1744

一个盒子中装有 ${\displaystyle 2}$ 个黑球和 ${\displaystyle 8}$ 个红球. 随机地从盒子中取一个球, 观察其颜色后放回, 若出现连续两次取到红球, 则停止取球, 那么取球总次数的数学期望为_____.

记 ${\displaystyle X_i}$ 为连续取到两次红球并停止取球, 且当前连续取到了 ${\displaystyle i}$ 次红球, ${\displaystyle i=0,1}$.
则 $$\{\begin{array}{rl}& \mathbb{E}(X_0)=\frac{1}{5}(\mathbb{E}(X_0)+1)+\frac{4}{5}(\mathbb{E}(X_1)+1),\\ & \mathbb{E}(X_1)=\frac{1}{5}(\mathbb{E}(X_0)+1)+\frac{4}{5},\end{array}$$ 解得 ${\displaystyle \mathbb{E}(X_0)=\frac{45}{16}}$.