素数骰：点数和为素数的概率

除了 2 外所有素数都是奇数，因此若两数都为奇数，它们的和为偶数且大于 2，不可能是素数。

所以必须有一个是 2。此时和为素数要求 $2+(\text{另一个})$ 为素数。

可行的素数和为 5、7、13，对应对偶为 (2,3)、(2,5)、(2,11)，每个有 2 种排列，共 6 个有利结果。

总结果 $6^2=36$，故概率为

$$\frac{6}{36}=\frac{1}{6}.$$

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Original Explanation

The numbers on the sides would be $2, 3, 5, 7, 11,$ and $13$. Therefore, the sum of the two upfaces has to be between $4$ and $26$, inclusive. The prime integers in this interval are $5, 7, 11, 13, 17, 19,$ and $23$. We now need to determine how each prime can be obtained from these dice. One thing to note is that $2$ must be one of the rolls, as all other values on the die are primes larger than $2$, which must be odd. Therefore, the outcomes are just primes $p$ such that $p-2$ is also a prime and $p-2 \leq 13$. The values of $p$ where this holds true is $p = 5, 7,$ and $13$. Each of these have two permutations of the die outcomes that yield that sum. Therefore, $6$ such outcomes of the $6^2 = 36$ yield a prime sum, so our answer is $\dfrac{6}{36} = \dfrac{1}{6}$