P5644 [PKUWC2018] Hunter Kill

Hunter Kill is a folk version of the once-popular game “Werewolf”. The rules are as follows. At the beginning, there are $n$ hunters. The $i$-th hunter has a hatred value $w_i$. Each hunter has only one fixed skill: after dying, they must fire one shot, and the person who is shot will also die. However, choosing whom to shoot is also important. Suppose the hunters still alive are $[i_1\ldots i_m]$. Then the probability of shooting hunter $i_k$ is $\frac{w_{i_k}}{\sum_{j = 1}^{m} w_{i_j}}$. At the beginning, the first shot is fired by you, and the target is chosen in the same way as the hunters (that is, hunter $i$ is hit with probability $\frac{w_i}{\sum_{j=1}^{n}w_j}$). Due to the chain reaction caused by shooting, all hunters will eventually die. Now hunter $1$ wants to know the probability that they are the last one to die. Output the answer modulo $998244353$.

The first line contains a positive integer $n$. The second line contains $n$ positive integers. The $i$-th positive integer denotes $w_i$.

Output the answer.

#### Sample Explanation The answer is $\frac{2}{4}\times \frac{1}{2}+\frac{1}{4}\times \frac{2}{3}=\frac{10}{24}$. For $10\%$ of the testdata, $1\leq n\leq 10$. For $30\%$ of the testdata, $1\leq n\leq 20$. For $50\%$ of the testdata, $1\leq \sum\limits_{i=1}^{n}w_i\leq 5000$. For another $10\%$ of the testdata, $1\leq w_i\leq 2$, and $w_1=1$. For another $10\%$ of the testdata, $1\leq w_i\leq 2$, and $w_1=2$. For $100\%$ of the testdata, $w_i>0$, and $1\leq \sum\limits_{i=1}^{n}w_i \leq 100000$. Translated by ChatGPT 5