I 3. A 10 metre long path consists of 20 square 1 metre by 1 metre flagstones laid in a 2 × 10 rectangular grid. You stand on the left hand flagstone at one end of the path and move among the flagstones making up the path randomly in discrete time steps. At each time step you pick a flagstone which shares an edge with the one you are standing on (so your move may be left, right, forward or back but not diagonal) uniformly at random and move on to it. (a) Describe a Markov chain which models this process. (b) Say why you chose to display the transition probabilities in the way you did in part (a)? (c) Using non-mathematical language, write down an example of a probability that could be calculated using first-step analysis. (d) Using non-mathematical language, write down an example of a random variable whose expectation could be calculated using first-step analysis.
I 3. A 10 metre long path consists of 20 square 1 metre by 1 metre flagstones laid in a 2 × 10 rectangular grid. You stand on the left hand flagstone at one end of the path and move among the flagstones making up the path randomly in discrete time steps. At each time step you pick a flagstone which shares an edge with the one you are standing on (so your move may be left, right, forward or back but not diagonal) uniformly at random and move on to it. (a) Describe a Markov chain which models this process. (b) Say why you chose to display the transition probabilities in the way you did in part (a)? (c) Using non-mathematical language, write down an example of a probability that could be calculated using first-step analysis. (d) Using non-mathematical language, write down an example of a random variable whose expectation could be calculated using first-step analysis.
ChapterP: Prerequisites
SectionP.6: The Rectangular Coordinate System And Graphs
Problem 37E: An airplane flies from Naples, Italy, in a straight line to Rome, Italy, which is 120 kilometers...
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3. A 10 metre long path consists of 20 square 1 metre by 1 metre flagstones laid in
a 2 × 10 rectangular grid. You stand on the left hand flagstone at one end of the
path and move among the flagstones making up the path randomly in discrete time
steps. At each time step you pick a flagstone which shares an edge with the one you
are standing on (so your move may be left, right, forward or back but not diagonal)
uniformly at random and move on to it.
(a) Describe a Markov chain which models this process.
(b)
Say why you chose to display the transition probabilities in the way you did in
part (a)?
(c) Using non-mathematical language, write down an example of a probability that
could be calculated using first-step analysis.
(d) Using non-mathematical language, write down an example of a random variable
whose expectation could be calculated using first-step analysis.
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