看遍所有点数

这是经典的 coupon collector（集齐卡券）问题。

对于 $n$ 个等概率结果，看到所有结果至少一次的期望试验次数是 $$E(n)=nH_n,$$ 其中 $$H_n=1+\frac12+\frac13+\cdots+\frac1n.$$

这里 $n=6$，所以 $$H_6=1+\frac12+\frac13+\frac14+\frac15+\frac16=\frac{49}{20}=2.45.$$

因此 $$E(6)=6\cdot H_6=6\cdot\frac{49}{20}=\frac{294}{20}=14.7.$$

答案是 $$\boxed{14.7}.$$

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Original Explanation

For $n$ equally likely outcomes, the expected number of trials to see all $n$ at least once is

$$E(n) = n \cdot H_n,$$ where $$H_n = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}.$$

For $n=6$: $$H_6 = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6}.$$

$$H_6 = \frac{49}{20} = 2.45.$$

Multiply by $n=6$ $$E(6) = 6 \cdot H_6 = 6 \cdot \frac{49}{20} = \frac{294}{20} = 14.7.$$

$$\boxed{14.7}$$