煎饼问题

答案为 $\boxed{\tfrac{2}{3}}$。

共有 3 张煎饼：GG、BB、GB。你看到“金色”意味着看到的是 6 个面中的 3 个金色面之一；其中 2 个金色面来自 GG（两面都金），1 个来自 GB。

在已看到金色的条件下，落在 GG 的概率为 $2/3$，此时另一面必金；落在 GB 的概率为 $1/3$，此时另一面为黑。

因此所求概率为 $2/3$。

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英文解析

Solution 1: Label the pancakes 0,1 , and 2 , according to the number of burnt sides. Let $E_{i},i = 0,1,2$ , denote the event that the pancake with $i$ burnt sides

is drawn from the bag. Let $A$ denote the event that the side (of the randomly drawn pancake) we look at is golden.

The probability of the pancake drawn from the bag having no burnt sides given that one side is golden is $P\left(E_0 \mid A\right)$ . Then, from the Bayes' formula, we obtain that

$$\begin{array}{l}{{P\left(E_{0}\mid A\right)=\frac{P\left(E_{0}\cap A\right)}{P(A)}=}}\\ {{\frac{P\left(E_{0}\right)P\left(A\mid E_{0}\right)}{P\left(A\mid E_{0}\right)P\left(E_{0}\right)+P\left(A\mid E_{1}\right)P\left(E_{1}\right)+P\left(A\mid E_{2}\right)P\left(E_{2}\right)}}}\\ {{=\frac{1}{1\cdot\frac{1}{3}+\frac{1}{2}\cdot\frac{1}{3}+0\cdot\frac{1}{3}}}}\\ {{=\frac{2}{3}.}}\end{array}$$

Solution 2: Out of the six possible sides that we could have seen, three are golden. Out of these, two belong to a pancake that is golden on both sides. Therefore, the probability of the other side being golden is $\frac{2}{3}$ .