1Markov ChainChapter 12Finding a cheese in the mazeCheeseBlack holemouse3AssumptionsThe mouse does not know where the cheese is.The mouse will get nothing and leaves the game if it enters the black hole.4QuestionsA biologist may want to know the following things.Can the mouse detect the cheese by smell?Can the mouse self-train if it is put back to the game after entering the black hole?Would the chance of getting cheese be the same for different initial position?
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5Establish a probabilistic modelTo determine if the mouse can find cheese by smell, we use the following idea.Determine the probability of success under the alternative hypothesis.Conduct experiments and compare the results with the probability.If the successful rate is not higher than the theoretical probability significantly, then the hypothesis is rejected. 6Establisha Probabilistic modelSelf-train hypothesis. Idea:Use special materials to make the maze so that the mouse cannot find cheese by smell.Determine the probability under the alternative hypothesis.If the successful rate is not higher than the theoretical probability significantly, then the hypothesis is rejected.7Model construction: ass. room #15141312111098765432108Model construction: stochasticDefine the stochastic variable Xnbe the position of a mouse at step n.X0= 0 means the mouse is initially placed in room 0.Successful probability: P(Xn= 13, for some n)Failure probability:P(Xn= 7, for some n)