Problem F
Arcade!
Write a program that computes the expected payout when dropping a ball into the machine!
Input
The input consists of a single test case. The first line contains an integer $N$ ($1 \le N \le 32$) describing the number of rows of the arcade machine. The second line contains $H = N (N+1) / 2$ integers $v_ i$ ($-100 \le v_ i \le 100$) describing the payout (positive or negative) if the ball drops into hole $i$. Holes are numbered such that hole $1$ is in the first row, holes $2$ and $3$ are in the second row, etc. The $k^{\textrm{th}}$ row starts with hole number $k (k-1) / 2 + 1$ and contains exactly $k$ holes.
These two lines are followed by $H$ lines, each of which contains $5$ real numbers $p_0 \ p_1 \ p_2 \ p_3 \ p_4$, denoting the probability that the ball bounces to its top-left ($p_0$), top-right ($p_1$), bottom-left ($p_2$), or bottom-right ($p_3$) neighbors or that the ball enters the hole ($p_4$). Each probability is given with at most $3$ decimal digits after the period. It is guaranteed that $0.0 \le p_ i \le 1.0$ and $\sum p_ i = 1.0$. If a hole does not have certain neighbors because it is located near the boundary of the arcade machine, the probability of bouncing to these non-existent neighbors is always zero. For instance, for hole number $1$, the probabilities to jump to the top-left and top-right neighbors are both given as $0.0$.
You can assume that after the ball has bounced $b$ times, the probability that it has not fallen into a hole is at most $(1 - 10^{-3})^{\lfloor b/H \rfloor }$.
Output
Output a single number, the expected value from playing one game. Your answer is considered correct if its absolute or relative error is less than $10^{-4}$.
Hint: Using Monte Carlo-style simulation (throwing many balls in the machine and simulating which hole they fall into using randomly generated choices) does not yield the required accuracy!
| Sample Input 1 | Sample Output 1 |
|---|---|
4 40 30 30 40 20 40 50 30 30 50 0.0 0.0 0.45 0.45 0.1 0.0 0.3 0.3 0.3 0.1 0.3 0.0 0.3 0.3 0.1 0.0 0.3 0.3 0.3 0.1 0.2 0.2 0.2 0.2 0.2 0.3 0.0 0.3 0.3 0.1 0.0 0.8 0.0 0.0 0.2 0.4 0.4 0.0 0.0 0.2 0.4 0.4 0.0 0.0 0.2 0.8 0.0 0.0 0.0 0.2 |
32.6405451448 |
| Sample Input 2 | Sample Output 2 |
|---|---|
2 100 50 50 0.0 0.0 0.45 0.45 0.1 0.0 0.90 0.0 0.0 0.10 0.90 0.0 0.0 0.0 0.10 |
76.31578947368 |
