挑剔的素数：选 4 个素数之和为偶数

除 2 外所有素数都是奇数。

4 个数之和为偶数当且仅当选到偶数个奇数；由于最多只能选到一个偶素数 2，若包含 2，则其余 3 个为奇数，和为奇数。

所以和为偶数等价于“没有选到 2”，即 4 个都从其余 15 个奇素数中选。

因此概率为

$$\frac{\binom{15}{4}}{\binom{16}{4}}=\frac{3}{4}.$$

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Original Explanation

All primes besides $2$ are odd, so we get a sum that is odd precisely when we select integers that are all not $2$. In other words, we select our $4$ integers from the other $15$ primes. There are $\binom{15}{4}$ ways to pick the $4$ from the other $15$ and $\binom{16}{4}$ total ways to pick $4$ primes from the $16$. Therefore, our probability is

$$\frac{\binom{15}{4}}{\binom{16}{4}} = \frac{3}{4}$$