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.
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.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Transcribed Image Text: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 THTHTTTHHHHTHHTHTTTT, 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?
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