问题描述
我有一个迷宫，如上所示(使用链接)，状态3包含奖赏，而状态7包含休克.鼠标可以随机置于1到9的任何状态，并且可以随机均匀地在迷宫中移动
I have a maze as shown above(use the link) and state 3 contains prize while state 7 contains shock. a mouse can be placed in any state from 1 to 9 randomly and it move through the maze uniformly at random
Pi表示在AIM在隔室i中启动的情况下，鼠标在状态7之前到达状态3的概率.
Pi denote the probability that mouse reaches state 3 before state 7, given that AIM started in compartment i.
如何计算∈{1,2,3,4,5,6,7,8,9}的Pi.
how to compute Pi for ∈ {1,2,3,4,5,6,7,8,9}.
推荐答案
让 Px 为游戏在 3 位置结束的概率> x .
Let Px be the probability that the game ends in position 3 if it starts in position x.
我们知道 P3 = 1 和 P7 = 0
如果您从任何其他单元格开始，那么在移动之后，您实际上将在新的单元格中再次开始游戏.因此，可以根据它们可以移动到的邻居的概率来计算其他7个单元的概率:
If you start in any other cell, then after you move you are essentially beginning the game again in the new cell. The probabilities for the other 7 cells can therefore be calculated from the probabilities for their neighbors that they can move to:
P5 = P2/4 + P4/4 + P6/4 + P8/4
P2 = P1/3 + P5/3 + P3/3
P1 = P2/2 + P4/2
...等等
对于每个单元格都有一个线性方程式-9个方程式对应9个单元格.使用高斯消元法或类似技术来求解这9个概率的方程组.
For each cell you have a linear equation -- 9 equations for 9 cells. Use Gaussian elimination or similar technique to solve the system of equations for the 9 probabilities.
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