Question

Imagine that you flip a fair coin 20 times. This will create a sequence of heads and tails, and there are lots of possible sequences (just over one million equally likely microstates). Now, imagine looking for strings of consecutive H's or consecutive T's within each sequence of 20 flips. For instance, one sequence (one microstate) is THHTTTTHTHHHTTHHTTHT, which contains one string of four consecutive T's, one string of 3 consecutive H's, and many strings of two consecutive H's and T's. Let's focus on the LONGEST string of consecutive H's or T's in each 20-coin sequence and define the LENGTH of the longest string to be the "macrostate" for that particular microstate. In the example above, the macrostate is "4", because the longest string is a string of four consecutive T's. (a) How many microstates are there for macrostate = 1? Explain. How many microstates are there for macrostate = 20? Explain. How many microstates are there for macrostate = 19? Explain. (b) What are the five most likely macrostates? What is the probability that each of these macrostates occurs? Part (a) can be done without any computational tools, and it helps to list the possible microstates as part of your explanation. However, part (b) is different. We cannot list every microstate (every possible sequence), because we cannot create a well functioning worksheet with over one million rows, so instead, we will simulate the results with a much smaller number of trials in order to estimate the answers to the questions in part (b). Submit your Excel workbook to Moodle to be graded, i.e., do NOT submit a PDF with values and formulas.

Imagine that you flip a fair coin 20 times. This will create a sequence of heads and tails, and there are lots of possible sequences (just over one million equally likely microstates).
Now, imagine looking for strings of consecutive H's or consecutive T's within each sequence of 20 flips. For instance, one sequence (one microstate) is THHTTTTHTHHHTTHHTTHT, which contains one string of four consecutive T's, one string of 3 consecutive H's, and many strings of two consecutive H's and T's.
Let's focus on the LONGEST string of consecutive H's or T's in each 20-coin sequence and define the LENGTH of the longest string to be the "macrostate" for that particular microstate. In the example above, the macrostate is "4", because the longest string is a string of four consecutive T's.
(a) How many microstates are there for macrostate = 1? Explain. How many microstates are there for macrostate = 20? Explain. How many microstates are there for macrostate = 19? Explain.
(b) What are the five most likely macrostates? What is the probability that each of these macrostates occurs?
Part (a) can be done without any computational tools, and it helps to list the possible microstates as part of your explanation. However, part (b) is different. We cannot list every microstate (every possible sequence), because we cannot create a well functioning worksheet with over one million rows, so instead, we will simulate the results with a much smaller number of trials in order to estimate the answers to the questions in part (b).
Submit your Excel workbook to Moodle to be graded, i.e., do NOT submit a PDF with values and formulas.

Added by Colton C.

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