Science:Math Exam Resources/Courses/MATH152/April 2010/Question B 03 (a)

MATH152 April 2010

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[hide] Question B 03 (a)
A random walk problem with two states has a transition matrix $P=\left[{\begin{array}{cc}1/3&1/4\\2/3&3/4\end{array}}\right]$ a) What is the probability that when the walker is in state 1 it moves to state 2?

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[show] Hint
What vector corresponds to the walker being at state 1 (with certainty 1)?

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[show] Solution

If the walker starts in state 1, then the probability vector is given by $\mathbf {v} =\left[{\begin{array}{c}1\\0\end{array}}\right]$ (i.e: the walker is in state 1 with probability 1 and in state 2 with probability 0). To compute the probability that the walker is in state 2 after one step, simply compute $P$v and look at the second entry: $P\mathbf {v} =\left[{\begin{array}{c}1/3\\2/3\end{array}}\right]$ Thus, the probability of switching from state 1 to state 2 in one time step is 2/3. (Equivalently, to answer this question you could look at the second entry in the first column of the matrix $P$).