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

MATH152 April 2010

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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?

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

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

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. 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 $P$).